Need advice, maybe help with ID

Hello,
I signed up here to ask this question since I wasn't really successful searching on the internet about this.

This was my grandfather's watch, and he's had it for as long as I can remember back, but I have no idea where or when he got it. I inherited it after he died 10 years ago, but it stayed in a drawer until I found it again now.
It started running immediately again when I took it out, and it goes about 15-20 seconds behind per day, which seems impressive to me considering the circumstances.

It's between 32 and 33 cm according to my measurement, and it looks to be gold, but I wouldn't know how to test it.

I can't open up the back unfortunately, because I don't want to scratch it, but identification isn't that important to me anyway (I'll take it to a watchmaker anyway), although I would be happy if you can tell what model/year it is just from the photos.
And I don't care about its value unless you think it's fake (which would be shocking to me, because it would be my grandfather's style).

I haven't found an image of a watch quite identical to mine on the internet (no date window, no special label, seconds hand in the center, no label, straight lug ends), but many similar ones from the early 60s, so I would assume it's from that era and a lower end model.

My actual question is what you think I should do with it. I would like to have a professional remove the dirt ring around the glass and the dirt around the indices and the omega logo inside the glass, because I think that's not very aesthetically pleasing. Do you think that's a good idea, or is that 'patina' that belongs on the watch? Is that even possible without replacing the whole dial (which I wouldn't want)?

replacing the whole dial (which I wouldn't want)?

Be careful how you proceed. The watch appears to be solid gold. A hallmark can be seen on one of the lugs and I suspect that will tell you what karat the gold is. In my opinion, it is not as you suspect a "lower end" watch. I will defer to members with more experience in matters of dials and their refurbishment. Go slow.

I think that it might have the case reference number 161/2 5002. It has a 522 movement. If this is so then it was produced in the mid 60s and is probably 9kt gold. If it is from the Uk then you might find another case reference number on the back, which means it was assembled in the UK by another company, Dennison.
As far as restoration goes, I think your best bet is to find a replacement dial, they are not difficult to find once you have the correct reference number. You will be surprised what a good clean up and a bit of tlc can do for a watch. Depending on where you are in the world someone on here can recommend a good watch maker to get it serviced and cleaned up. It will probably cost you more than the watch is worth but that is not relevant as it is a reminder of your grandfather.

I'm in Vienna, I doubt the watch has any connection to the UK.

But thank you very much for the info.

first thing to do is to open it to see the number and second thing is to have it serviced. It's probably a 552 inside, which is a good movement