Omega 1250 cal - quartz replacement?

Replacing the movement a quartz movement is a bit on a long shot, imo.

I successfully serviced an Accutron 214, which is a similar predecessor to your Omega 1250. My experience was that it requires a very clean index wheel with no damage. Did you verify yours under a good microscope? Once you verified the index, the pawls need to be adjusted under microscope for alignment and proper phasing.

The best technical guide I've found is for the Longines cal. 6312 (really an ESA 9162 like the Omega cal. 1250) but that manual, same as the Omega and ESA ones, doesn't provide dimensions. Assuming you're okay using temporary hands to go along with a temporary movement, the critical dimensions will be date position and stem height. Since you have the movement, you can measure those. My guess would be that the stem height will be significantly more than common quartz movements, but that's just a guess. No idea about the date position, but maybe you could live with an empty placeholder there temporarily.

One thing that probably isn't a concern is movement size. Esslinger sells an inexpensive collection of plastic movement holders that are made to be cut to fit. There are like 50 in the package, and surely one will fit whatever quartz movement you find. You just cut the plastic to fit the watch case. It's not necessarily a long-term solution, but will work fine for your purposes.

As others have said, it's probably quite a long shot, but you might get lucky.