I know you can buy many types of pastes, papers and other wizardry for this. Like you I enjoy refreshing beaten-up old watches and try not to spend too much on equipment so -
For 'rough' finishing I use the same as what I use for polishing steel - 800 grit -> 1200 grit -> 2500 grit wet and dry (carbide) paper. Easily available although I have to go to Halfords (a car parts superstore) for the 2500 grit. Only use it wet on glass. I have a polished granite surface which I 'stick' the paper to and the rub the glass - keeps everything flat and avoids rounding the perimeter edges which is not cool. Then I finish with 3 micron cerium oxide (cheap little bag off Amazon.co.uk). The cerium needs to be in a paste or slurry to work. I have a cloth wheel for my not-Dremmel which seems to hold up better than the felt ones. I have soaked this in a wet slurry of cerium oxide to 'infuse' it with a particulated pinkness. Then I apply a paste to the surface and slowly polish it. Low speed - long-time is better for this - and stop for a while if it gets too hot. I can't fill the case with water because I don't do it upside down but you could if you have a bench mounted wheel. I just hold the case with my thumb on the backside of the glass under where I'm polishing as an over-temp sensor.
When your watch glass is wet you obviously cannot see the pregress of scratch removal. I find coffee filters are handy for quickly drying the face to inspect progress.
Whatever you use be scrupulously clean and segregate all of your grits and powders or you risk wasting a lot of time with mysterious scratches appearing during final polishing. Clean off the glass well before moving down a grit size. Keep any 'infused' wheels cleanly stored for reuse only with that grit size. If your chosen paper has any folds, tears or areas of wear in the area you are using then use it for something else and get a new piece - one piece of detached grit will quickly spoil your day unlike it will on steel.
This method works well on 'older' mineral glasses although I must say it took a very very long time on my modern Citizen which (not sapphire) has some kind of unholy Mineral glass that will take a scratch quite happily from little knocks on everyday items but apparently doesn't very much like being re-polished.
Rgds,
MP.