Very nice watch.
It looks good to me, but I think the bracelet was either a customisation choice at time of purchase, or has been subsequently replaced. I think most likely it was on display with a leather strap and the customer wanted a gold bracelet (more about that below).
The movement markings look fine and 13210896, would mean it was made sometime around 1952 (serial numbers between 13000000-13999999). Some of the serial number ranges span several years, so dating can only be an approximation.
There’s no reason why the serial number on the case should match the movement since the case was made for, not by, Omega. The marks on the case all look good for Swiss hallmarked 18k gold and the ‘key of Geneva’ to the right designates that as the assay office. The number in the key is the registration number for the maker, and ‘29’ is for Weber & Cie, SA of Geneva. ‘Cie’ is short for ‘Companie’ (Company) and ‘SA’ is for ‘Soci?t? Anonyme’ (the Swiss equivalent of Public Limited Company). They first registered in 1934 and then again in 1938 as a Limited Company.
The bracelet tells a different story I think. That hexagon mark on the clasp could be read either way up but it looks to me like a Spanish hallmark, which would have a number at the top and a letter or pair of letters below. It looks to me like ‘69’ above the letter ‘B’. If I read it correctly that would be for the assay office in Barcelona, Spain and the numbers would be the maker’s registration number… but I can’t immediately put my hands on a full list of maker numbers.