Somebody here has found a replacement battery, that worked well. But I can't remember any details. I'll do some searching and get back if I find anything. But in the meantime good luck cleaning the terminals, that's any easy fix, if the battery is still OK.
Well that didn't take long, occasionally I do the sensible thing and take notes.
That's 602025 for the new. 552425 for the old.
The new is 6mm thick 20mm wide and 25mm long.
The old is 5.5mm thick 24mm wide and 25mm long
So google 602025 lithium battery, it's not quite the same but it fits.

See how the old one has blown up. If you open the back of the watch and your battery looks like that, you need the replacement
I thought I had some pics of the GW60 opened up, but so far I can't find them.
OK, traps for old farts, store your phone pics under "phone pics". look for GW60 pics under GW60. hallelujah for the search function.

so looks like a fine tipped soldering iron is required to change the battery