Please take this advice as you will - I'm not a professional watch restorer, just a private collector. Gentle cleaning solutions will do a lot to lift the "grubbiness" of a vintage watch. I'm not calling your watch grubby, I just can't think of a better word for this sort of thing.
If you don't have access to a sonic cleaner, I have used two odd products that I have to hand which work well: Optrex and 3M surface cleaner. Remove the bracelet and drop that into a shallow bowl of Optrex liquid - leave it there for half an hour and you will be amazed at how much dirt simply comes out. Then, get a Q-tip (other generic cotton buds are available) and work between the links. That stuff gets out everything from general dirt to mineral oil. Pat dry thoroughly with kitchen paper when you're done. 3M surface cleaner is harder to get hold of and much more expensive, but it is a specialist cleaner that leaves no residue and can be used on all materials - so it's not going to eat away at seals etc. Just use a little around the glass with a Q-tip again.
Finally, if you want to give an actual gentle polish, Cape Cod cloths are pretty damn good. They won't take out deep scratches but they'll certainly enhance any minor "bruises" and leave a shiny finish. For acrylic glass, polywatch does a great job too. I had a fish around for any before and after photos of anything I had and there's an old (and very poorly) Sorna that I picked up on ebay below - you can particularly see the difference that 15 minutes with a little polywatch makes to the glass. The case was just treated with a little Optrex.
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