Let's see if we can narrow some of the options down and help you in trying to find a replacement.

Square pushers and rounded crown instantly makes me think 40s rather than just before the Carrera. At the time, most of the production was available in solid gold as well as steel too.

Something like the watches on this page:

Can you remember anything about the Heuer shield itself? At the time of the catalogue shot above, they were still tending to use the shallow version of the shield. Later in the 40s and into the 50s, the dimensions change to be like the shield as we know it today, so that could give us another piece of dating evidence.

Jumping forward 15 years or so, we can look at some of the pre-Carreras. Some of these were in gold and did look strongly like the Carrera (indeed, occasionally they end up for sale with Carrera added to the dial. Check the 415 on this page:

In that case, though, we would definitely be talking about round pushers and dauphine hands. The 3641 did later go on to use Carrera hands and dial markers, but I don't recall these "economy" models ever having solid gold cases. It would seem to defy reason to do that, but having worked through piles of Carreras, I csn say that rhyme and reason were sometimes at a premium where Heuer were concerned!

So I guess the next step is a question for you - which are the strongest memories of the watch? The square pushers and rounded protruding crown, in which case we can focus on the 40s and 50s, or the very Carrera-esque dial, which means looking at the 60s?