Although there are various designs of lathes intended for use by watchmakers, some dating back to the late 1700s and including specialised models - for example "fiddle" lathes,  "steel turns", Jacot, Swiss, Swiss Universal (also called the English Mandrel) Bottum and Dracip - more modern examples can generally be divided into two types: the lighter "Geneva" and  heavier "WW". The "Geneva"  version can be recognised by its round bed, with a flat machined along either the top (or, more commonly, the back) and nearly always supported on a single foot beneath the headstock. These lathes, invented in 1859 by Charles S. Moseley in the U.S.A., generally take 6, 6.5 or 8 mm collets and were designed for lighter, very high-precision work. However, the "WW" (Webster-Whitcombe), is considered by many to be a more popular and versatile machine and is also of American origin, dating from around 1889. The centre height of the WW is usually 50 mm, though very occasionally 65, 70 mm and other figures are encountered. Of heavy construction, the bed is formed with a 37 mm-wide flat on the top and a 60-degree bevel along each edge, and carries a headstock spindle to accept 8, 10 mm or 12 mm collets (though odd, in-between sizes have also been discovered).
Amongst exceptions to the above two types are a number of unusual and interesting machines including a range of fine Boley lathes, manufactured from the middle 1800s to the early 1900s with either a triangular-form bed or an oddly compromise design (which did not catch on) where the bottom of the bed was semi-circular in form and the top triangulated or "bevelled" . Also available have been slightly larger lathes with triangular-beds such as Glashutte, H.Strube & Fils together with unknown makes from the former East Germany and, though they might be considered as being too large, an unknown model from France and the Dalgety.
Larger than the WW type are what might be called "toolmakers' or "bench precision" lathes: these vary in size from the Schaublin 65 and 70 (the latter being the most popular and frequently-encountered machine in the professional watchmaker's workshop), the Pultra 15/90 and larger examples such as the Schaublin 102, American Watch Tool Company, Arrow, B.C.Ames, Bausch & Lomb, Benson, Boley, Bottum, Boxford, B.W.C., Carstens, Cataract, Cromwell, Crystal Lakes, CVA, Derbyshire, Elgin, Hardinge, Hjorth, Juvenia, Karger, Leinen, Levin, Lorch, Mikron, W.H.Nichols, Potter, Pratt & Whitney, Rambold, Rebmann, Remington, Rivett, Saupe, See (FSB), Sloan & Chace, Smart & Brown, T & L.M., U.N.D., Van Norman, Wade, Waltham Machine Works, Weisser, Wolf Jahn and (though now very rare), Frederick Pearce, Ballou & Whitcombe, Sawyer Watch Tool Co., Engineering Appliances, Fenn-Sadler and the "Cosa Corporation of New York." While useful machines in a precision workshop, these are outside the scope of this article but further information can be found here.
There have been not only dozens of makers of watchmakers' lathes but also a lot of "badge engineering" where dealers and distributors commissioned batches of lathes for themselves, a well-known example in England being George Adams of London who re-sold mainly German-made machines and G.Boley who made available their "bevelled-bed" model for re-branding, two examples being the "English" Rawco and Telco. This situation was compounded by a number of parts and accessories being interchangeable between different makers - so it is entirely possible that a lathe has been "made up" from others, either casually over many years of ownership, or unscrupulously by a "dealer". However, it's very unlikely that the bed, headstock and tailstock will be from different manufacturers, but if they are, be wary.
Unfortunately no maker of a watch lathes has ever offered a proper handbook for their products but, happily, there is an excellent hard-back book available that does the same job: "The Watchmakers' Lathe".  This is a long-established publication and, because most of these lathes were built along the same lines, and use almost identical accessories, the book is able to give instructions and guidance that applies to all types.
Below is a set of links in the Machine Tool Archive of lathes.co.uk to some makers of watch and instrument makers' lathes and their accessories - some rather common, such as G.Boley, and others very rare, like the O.W.T. and Reese.

A useful link for collets associated with watchmakers' lathes is
http://www.lathes.co.uk/collets

http://www.lathes.co.uk/Aero
http://www.lathes.co.uk/American Watch Tool Company
http://www.lathes.co.uk/Andra & Zwingenberger
http://www.lathes.co.uk/ARS
http://www.lathes.co.uk/Boley/page3.html
http://www.lathes.co.uk/Boley/page4.html
http://www.lathes.co.uk/Boley/page5.html
http://www.lathes.co.uk/Boley/page6.html
http://www.lathes.co.uk/Boley/page7.html
http://www.lathes.co.uk/Bergeon
http://www.lathes.co.uk/Bottum
http://www.lathes.co.uk/BTM
http://www.lathes.co.uk/chinese-watch-lathe
http://www.lathes.co.uk/Clement
http://www.lathes.co.uk/CLH
http://www.lathes.co.uk/Coronet-Precision
http://www.lathes.co.uk/Cowells
http://www.lathes.co.uk/crown
http://www.lathes.co.uk/Derbyshire
http://www.lathes.co.uk/East-German (Unknown makes)
http://www.lathes.co.uk/English-Mandrel
http://www.lathes.co.uk/Favorite
http://www.lathes.co.uk/elson
http://www.lathes.co.uk/Gamma
http://www.lathes.co.uk/GemGlorious
http://www.lathes.co.uk/George-Adams
http://www.lathes.co.uk/Glashutte
http://www.lathes.co.uk/Go-Ahead
http://www.lathes.co.uk/Hammel-Riglander
http://www.lathes.co.uk/Hazemeyer
http://www.lathes.co.uk/Hopkins
http://www.lathes.co.uk/IME
http://www.lathes.co.uk/Rudolf-Kadner
http://www.lathes.co.uk/Kampe
http://www.lathes.co.uk/Kearney & Swart
http://www.lathes.co.uk/Kharkovski
http://www.lathes.co.uk/Lanco
http://www.lathes.co.uk/Leinen-traditional
http://www.lathes.co.uk/Leinen-modern
http://www.lathes.co.uk/Levin
http://www.lathes.co.uk/lorchtriumph
http://www.lathes.co.uk/Lorch-Watchmaker
http://www.lathes.co.uk/Lorch
http://www.lathes.co.uk/Mahora
http://www.lathes.co.uk/Marco
http://www.lathes.co.uk/Marshall
http://www.lathes.co.uk/Moseley
http://www.lathes.co.uk/Nordan
http://www.lathes.co.u/OWT
http://www.lathes.co.uk/Paulson
http://www.lathes.co.uk/Peerless
http://www.lathes.co.uk/Pennant
http://www.lathes.co.uk/Perfection
http://www.lathes.co.uk/Perton
http://www.lathes.co.uk/Pioneer
http://www.lathes.co.uk/Pultra
http://www.lathes.co.uk/rawco
http://www.lathes.co.uk/Riken
http://www.lathes.co.uk/Rivett
http://www.lathes.co.u/ROFB
http://www.lathes.co.uk/Rolls-Royce
http://www.lathes.co.uk/russian-watchmakers
http://www.lathes.co.uk/Saacke
http://www.lathes.co.uk/Scomea
http://www.lathes.co.uk/Star
http://www.lathes.co.uk/Stark-watchmaker
http://www.lathes.co.uk/stehman-jenks-stehman
http://www.lathes.co.uk/Steiner
http://www.lathes.co.uk/superior
http://www.lathes.co.uk/Swiss-Universal
http://www.lathes.co.uk/Telco  (Boley)
http://www.lathes.co.uk/Wolf-Jahn
http://www.lathes.co.uk/Van-Norman
http://www.lathes.co.uk/ZDZ
http://www.lathes.co.uk/Zimmerman
……….. and others here

1. Headstock Spindle              2. Throat pin                        3. Loose bearing            4. Loose bearing pin
5. Adjusting nut                      6. Front bushing                   7. Rear Bushing             8. Front inside shield
9. Rear inside shield               10. Front outside shield       11. Rear outside shield
12. Pulley                                13. Pulley Hub                    14. Pulley screw            15. Draw-in spindle
16. Draw-in spindle wheel      17. Frame                            18. Index pin                  19. Bolt
20. Spring                                21. Eccentric                       22. Lever                        23. Pointed Centre
24. Spindle                               25. Spindle Button             26. Spindle Binder         27. Frame
28. Bolt                                    29. Spring                           30. Eccentric                  31. Lever
32. Slide                                   33. Pivot Screw                   34. Pivot Screw             35. Post
36. Lever                                  37. T graver rest                  38. Shoe                         39. Shoe bolt
40. Bolt pin                              41. Bolt washer                   42. Bolt spring               43. Bolt nut
44. Bed                                    45. Base                               46. Base bolt                  47. Bolt washer
48. Ball nut

An English-made BTM "Geneva" pattern watchmakers' lathe (note the bar-like bed with, in this case, a flat at the back) complete in its maker's fitted wooden box with a range of useful accessories. The item top left in the box is that most valuable of extras - a compound rest, used to hold a cutting tool manipulated under the control of screw-feed slides.

Boley Lathe No. 1b. A "WW" pattern lathe of a heavier pattern, admitting 230 mm between centres, and  with feet at both end of the bed. On this model the compound slide was fitted as part of the standard equipment.

Another Boley, but this time equipped for production work with quick-action levers fitted to the collet closer, compound slide and tailstock. The tailstock also has an indexing 6-station toolholder and the cross slide is fitted with toolposts at the front and the back.

A later type of  Boley & Leinen "Reform" lathe, manufactured from the early 1950s. The mouth-watering outfit was listed as the No. 77/X1V and contains virtually everything a skilled watch-repair man could wish for including, top left, a second headstock fitted with a permanently mounted (and hence very accurately aligned) faceplate with three clamps.

Continued:
Availability
Watchmakers" or Instrument-makers' lathes can be very valuable, especially if they are in fine, original condition and complete with lots of accessories. Buying just a basic lathe with bed, headstock tailstock and T-type tool rest can be false economy - there are lots of these about, at attractively low prices, but the real value is in the extras that allow the lathe to be used as a miniature "machining centre" - as originally intended - to cut, for example, wheels (gears), mill small parts and hold tiny, awkwardly shaped mechanisms for repair and restoration. Because there is considerable competition for accessories, if your basic lathe has to be equipped one part at a time much time, effort and money will be expended - hence, if you can, find a fully-equipped lathe; this will be a much better investment and a lot easier for you, or your heirs, to sell.
If you are looking for one of these machine I would strong recommend advertising for one. There are thousands sitting unused whose owners will never get round to advertising them - a "wanted" advertisement might just encourage them to get in touch:
Accessories:
Typically, the most valuable watchmakers' or instrument-makers' lathe would still be in its original wooden box with a wide range of equipment including as many of the following as possible:
Compound slide rest - screw-feed or lever action
Collets - a set of around 20 "Wire" (often called "split chucks"),
Collets - "Wheel" type in a set of 5 or 6
Collets - "Ring Step" type in a set of 5 or 6
Box Chuck
Chuck conventional 3-jaw Self-centring  (sometimes called a "Universal Chuck") in ring-scroll (knurled ring round the outside) and key-operated models,
Chuck conventional 4-jaw Independent
Cutting tools - as large a collection as possible
The following "chucks" mounted on collets:
Chucks - balance
Chucks - box type with screws through the body to hold jobs
Chucks - brass split type (sometimes called jewel type) to fit inside larger steel collets
Chucks - button or crown usually in sets of 10
Chucks - carrier for driving work between centres
Chucks - circular-saw type
Chucks - emery wheel
Chucks - lantern in bronze or steel,
Chucks - wax
Chucks - wood screw
Chucks - wood turning
Compound Slide Rest
Drill chuck for headstock or tailstock use
Drilling plates - self-centring (a disc with a ring of holes each formed with a coned face to self-centre work)
Drive Plate
Eye glass on adjustable holder
Fixed steady
Jacot Drum
Lapping attachment
Pivoting attachment
Saw table
Sinking tools
"Mandrel" - this has the appearance of a spare headstock with a "faceplate" attached and is used for super-precision work
Micrometer-adjustable boring head
Milling and Grinding Spindle,
Pivot polisher
Pivoting attachment
Filing rest in single or double-roller types
Rose cutters
Screwcutting Attachment with a set of changewheels
Sinking tools
T-rest - the basic device to rest a tool against. Available in standard and tip-over types
Tailstock chucks - also known as "drill stocks" and available with flat heads, V-heads and chuck type
Turning arbors
Topping or "rounding up" tool
Two types of Tailstock (sliding spindle and a lever-feed spindle),
Tip-over or simple sliding T-shaped Hand-rest,
Universal Faceplate and Pump Centre,
Vertical milling slide,
Wheel-cutting attachment with division plate (to cut what the laymen would call cogs or gears but which are known to the watchmaker as "wheels").
Drive systems
Even when fully equipped it is not unusual to find that a watchmaker's lathe has no drive system or even motor. However, this is rarely a problem for the easiest and cheapest solution is to use either the motor from a sewing machine or, preferably, a proper "Parvalux" unit - the latter available in 1-phase, 3-phase and DC types with speed ranges spanning 0.2 to 10,000 r.p.m. The motor can be bolted in place behind the headstock and driven by a special Swiss-made round belt that can be flipped easily from groove to grove, there being no need to make up the type of hinged countershaft that a larger lathe would need. The writer can supply Parvalux motors their controller and the special belting
Handbooks
Unfortunately no maker of a watch lathes has ever offered a proper handbook for their products but, happily, there is an excellent hard-back book available that does the same job: "The Watchmakers' Lathe".  This is a long-established publication and, because most of these lathes were built along the same lines, and use almost identical accessories, the book is able to give precise instructions that apply to all types..

A Swiss-made Bergeon Model C of the "Geneva" type. This example is fitted with a compound slide rest to hold the cutting tool, a lever-action tailstock and, to assist with very small work, a magnifying glass held on an articulated arm. Bergeon is the only indigenous Swiss manufacture of watchmakers' lathes to have survived into the 21st century.