First of all, welcome to the forum.
It is not correct to refer to battery connections as "live" and "neutral", these terms should be used in referring to mains supplies. (There are exceptions but lets keep it simple for now).
Just refer to the positive (+ve) and negative (-ve) when referring to batteries.
Just by coincidence, before logging on the the forum just now I was testing a couple of AA batteries, a short circuit test showed that they could deliver 6amps, that can make thin wires quite warm!.
So what was damaged in your case, probably not much.
The batteries will have lost some of their usefull life but are probably quite servicable for a long time yet.
The wire had its insulation melted, not the end of the world, just trim it off wirh a knife or side-cutters.
The battery holders, are they all melted out of shape? probably not, are the batteries still making good contact? yes they probably are.
So, no problem there then.
Another bit of pedantic terminology, strictly speaking one individual AA "battery" is really a cell, not a battery.
Only when you connect two or more cells together do they form a battery.
But dont worry, everybody uses the term battery for a cell.
Last of all, see my signature line. Appropriate in this case maybe.
JimB