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Some people here like Alsal. But I wouldn’t be scared to post it.
You’ll need the dial off to access the keyless works (which are at fault), so may as well get it serviced. I’d aim to not pay more than £125. You may need to buy a spare part for the keyless works (set lever spring or setting bridge) but that should only be around £10.
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I just recommended this guy a few posts up:
http://watchguy.co.uk/You'd need to post it though - unless you live near them. Seem super honest from my dealings with them.
I’ve asked this before “on here” but since forgotten. I have a Vertex Allproof that belonged to my grandfather. It has 1954 engraved on it so I guess that dates it... I think it has a slight issue, where the mechanism for winding it can get confused with that for actually setting the time.
Where is a good place to get this serviced (if that’s what is needed) in London. For reference it looks the same as this except the second hand doesn’t have a red arrow:
https://www.poshtime.com/3029.019/Vertex-Allproof-9ct-Gold-c.1953.php