Earth wire burnt out

Sounds like a short on the live side somewhere. Could be a bared wire or something like the Zener failing. PO replacing fuse with wire is a bad sign, and if there had been a fuse it may have saved you some grief.

Battery bulging sounds as though the short has damaged it by drawing excess current.

I'd be looking at checking all components and replacing the loom. And fitting proper fuses.....

When I started to read this problem about burnt out earth wire I recalled a MKIII fault like this years ago, but although 'my fault' burnt out all the loome earths it didn't 'bulge' the battery. Burnt out earths on MKIII was due to owner/restorer forgetting to earth the engine to the frame/battery with the high current thick cable so starter current used the loom earths! Frying tonight, was the result.

But the above fault looks like a real short that bulges the battery. I have had the original small power socket being the cause of a short as the owner put the self tapper screw through the electrical terminal instead of the mounting hole. Another Commando had an intermittent short/blown fuse but only when braking and going over bumps. Turned out to be the BROWN wire underneath the rear stop light switch that feeds the stop light was shorting on the stand-'sometimes' so 'sometimes' blew the fuse. Awkward one to find.

But all this is off topic so apologies, hope you find the short.

A short circuit result in far greater current than an alternator can give. The fuse ( which should have been there ) don't blow when the alternator gives all it can.

Could bet a substantial sum on a short circuit. Tricky thing is that the original cause can heat up some other wires so that insulation on them melts and creates new shorts. Careful inspection of the whole wiring is needed.

Mike