Manufacturers and Brands
Should any reader have a lathe from those below marked with the maker's name shown in red, the writer would be pleased to hear from you. Perhaps an example of each will, in the fullness of time, be found
Genuinely high-quality Watchmakers' lathes were manufactured and branded by, amongst others:
Accuro
Adams George (UK)
AF (Albert Froidevaux - Switzerland)
American Watch Company (C. S. Moseley-designed lathe circa 1859)
American Watch Company (A. Webster-designed lathe circa 1859/60)
American Watch Tool Company (Webster-Whiitcomb improved-design lathe of 1889 - the WW model)
Ames ((USA)
ARS (France)
J.A.Bradshaw (possibly just a dealer's marking)
Bergeon (Swiss)
Boley (German)
Boley-Leinen (German)
Boston Watch Company (C. S. Moseley-designed lathe circa 1858)
Bottum (USA)
Bourke (Australia)
Boydon
B.T.M. (UK)
Cardinali (Argentina?)
Cataract (Hardinge - USA)
C.L.H. (UK)
Coronet (UK)
R.Cowles ((USA) - possibly only a patent and never manufactured)
Dale
Derbyshire (USA)
Dracup
E.H.J. (E. H. Jones machinery dealers and commissioners UK)
Elson ((USA))
E.M.E.
Faneuil Watch Tool Co. (Rivett)
Favorite (Swiss)
Gem (Gem Glorious)
Gentil (Star Lathes, Switzerland)
George Adams (re-branded Boley and Lorch, etc. and cheaper imitations under his own label)
Hardinge (Cataract) (USA)
Hammel, Riglander & Co. (Europe)
Haskett ((USA) - possibly only a patent and never manufactured)
Hauser
Hazemeyer (Holland)
Hopkins - Patented 1872 (Van Norman - USA)
Horia (Swiss)
IME (UK)
Georg JACOB (Germany)
Jackson ((USA) - possibly only a patent and never manufactured)
R.H. St. John ((USA) - actually a specialised "dental lathe"
Jones (J & T Jones - UK)
J.E.Kampe (U.S.A.)
Gustav Klett (East Germany)
Lancaster Special (by the Faneuil Watch Tool Company circa 1899 for sale by a jewellers' supply house L. C. Reisner of Lancaster, Pennsylvania.
Lampert - (USA) - also a manufacturer of collets
Lancaster Special (USA)
Lanco (Lane Cove - Australia)
Leffel & Mulholland ((USA) - possibly only a patent and never manufactured)
Leinen (German)
Levin ((USA))
Lorch (Lorch Schmidt)
Luthy (Switzerland)
Manhora (French)
Mansfield
Marshall (USA)
Moseley (USA)
Nordan (Denmark)
Olin (Charles Olin of O.W.T. - USA)
O.W.T. (Ohio Watch Tool Co.)
Panerai & Figli (Italy)
Paulson (USA)
Peerless (USA)
Pennant (USA ?)
Perton (USA)
Prazima (GDR)
Precista
Picard (Henri Picard et Frere) - probably only a dealer
Pultra (UK)
Reese Thos. Haulton (USA)
Reliance
Rivett (sometimes marked Rivett Lathe Mfg. Co. and, possibly, Rivett Lathe & Grinder Co.)
Rivett Lathe & Grinder Corporation Type 1R
R.W. Roberts (UK)
ROFB (Royal Ordnance Factory, Birmingham, UK)
Saacke
Satif
Schmidt
Schaublin (Switzerland)
Schumer (Friedrich Schumer) Germany
Scomea (Société Commerciale d'Outillage et de Mécanique d'Aviation - France)
Shaller ((USA) - possibly only a patent and never manufactured)
Simplex
Skarzysko
Sloan & Chace (USA)
Star
Stark-watchmaker
Stehman-Jenks-Stehman
Steiner
Superior
Swartchild (agents)
Swan
"Swiss-Universal"
Szalay
T.C.M.
Taihang Instrument Factory Manufacturing
Taylor
Tribby ((USA) - possibly only a patent and never manufactured)
Unitor (Italy)
Vector (Germany) A copy built in China as well
Waltham (USA)
Webster (USA)
Webster-Whitcombe (WW)
Whitcombe (USA)
Wiskum (USA)
D.M. Williams
C.V.Woerd ((USA) - possibly only a patent and never manufactured)
Wolf, Jahn (Germany)
Van Norman (sometimes branded Hopkins) USA
ZDZ (Poland)
Zimmerman (USA)
Continued below:

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 toolposts at front and back.

A later type of  Boley & Leinen "Reform" lathe, manufactured from the early 1950s. This 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 or "dogs".

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: (many of those below are illustrated here)
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 (or triple) with 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
Chuck - conventional (but high-precision) 3-jaw or 6-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
Except where they are special-purpose, today "chucks" are usually called collets:
Chucks - "wire" for holding round material
Chucks - "fir tree" or "ring" used for holding the inside diameter of jobs
Chucks - "step" or "disc" used for holding the outside diameter of jobs
Chucks - "box" with four or more screws through the body to hold jobs
Chucks - "brass split" (sometimes called the jewel type) to fit inside larger steel collets
Chucks - "button" or "crown" usually in sets of 10 for holding watch winders
Chucks - "carrier" for driving work between centres
Chucks - circular-saw type
Chucks - emery wheel
Chucks - "lantern" in bronze or steel,
Chucks - "wax" used for holding parts using hot wax and other glues
Chucks - wood screw for turning wood and ivory, etc.
Chucks - blank end for the user to adapt as necessary for special jobs
Chucks - "step-up" for holding larger diameters
Chucks - balance, for holding the balance wheels used in watches
Cutting tools - as large a selection as possible
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 or catch plate  - for turning work between centres
Drive dogs - fastened to the work and "caught" by the catch plate and so turned with the spindle
Dividing attachment - mounted on the headstock spindle or a milling slide
Eye glass on adjustable holder
Filing rest in single or double-roller types
Fixed steady and travelling stead (the latter rare on watch lathes)
Jacot Drum
Lapping attachment
"Mandrel" - this has the appearance of a spare headstock with a "faceplate" attached and, as it is very accurately made in one piece, used for super-precision work
Micrometer-adjustable boring head
Milling slide
Pivoting attachment
Pivot polisher
Rose cutters
Screwcutting Attachment with a set of changewheels
Saw table - either a simple type to fit in the T-rest holder or a more complex universal version for mounting on the lathe bed
Sinking tools
Spindle - high-speed grinding and miller-cutter holding spindle--often attached to a vertical slide
T-rest - the basic device to rest a tool against. Available in standard and tip-over types and with different lengths of T
Tailstock chucks - also known as "drill stocks" and available with flat heads, V-heads and chuck type
Turning arbors - finely made bars to hold work. Can also be had in sets with very fine tapers of various diameters onto which jobs bored with a hole can be pressed
Topping or "rounding up" tool
Tailstock; many types including the basic sliding spindle and screw and lever-feed types - but also ones adapted for special work and, equipped with a capstan attachment, for production work.
Tip-over or simple sliding T-shaped Hand-rest,
Universal Faceplate and Pump Centre,
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 second-hand watchmaker's lathe has no drive system or even a 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, high-quality  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.
Handbooks
Amazingly, no maker of a watch lathes has ever offered a proper instructional handbook for their products. 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 manufacturer of watchmakers' lathes to have survived into the 21st century.