UPDATE:

I got a new battery today and the bike works. I found a local mechanic (here in Costa Rica) and he got hold of a battery for me. It's definitely a 12V battery and the same size as the previous one, but it has no markings on it. I have no idea if it is a "gel" one like the other one was or if that even matters. It worked today when we put it in, I've been for a spin and will try it out again tomorrow.

Good to hear it was an easy fix.