Please help identify this watch

The 9ct hallmarks suggest an English (or possibly commonwealth) origin for the case.
In which case you won’t find the usual Omega references inside the case back.
The movt serial number will give an approximate date of production for the movt and the hallmarks inside should give a year-date for the case - but you already know that from your father’s birthday.

Having the watch serviced is a really good idea - by an independent watchmaker used to vintage watches.
You could try @Mitka, a member here on OF.

And you could use a jeweller’s cloth to gently remove the tarnish.

Unless you’re really keen on the plated bracelet you could just buy a new strap. Those old plated bracelets can damage the lugs.

Looks like a UK market model 13308 which is a Dennison cased version of a Swiss model I can't quite place the number for right now. Or if not that one, some other similar Dennison model. It should have a hallmark (prob the Birmingham Anchor) and a date letter stamped on the inner caseback. If so, the letters changed on July 1st not Jan 1st in this era so bear that in mind for dating. The B'ham letter for late 1955/early '56 was capital F.

I can't tell from your pics, but if that bracelet, which is not original Omega btw, has spring loaded ends then ditch it pronto as it will be gouging circles onto the inner faces the lugs. Of course if your father wore it on that bracelet then perhaps that factor outweighs any minor damage!

If it is the model I suspect, it’s a full rotor not a bumper but he should be able to tell pretty easily of course!