If a new battery doesn't work, check this thread
http://www.talkphotography.co.uk/threads/olympus-om1-service.513937/
It is getting some charge with the current battery - the needle moves if the shutter speed is reduced to 1/4th and slower. Outside it actually gave a correct exposure reading at 1/1, f1.8!
Is this fix going to be tricky and/or expensive (if you've had experience of a similar problem)?
That's odd, clearly it was getting some current with the previous battery - but nothing with a new SR44?
I'd imagine it might be something to do with corrosion of the battery terminal wires, especially if a battery has been sitting inside for years.

So some current with the previous battery indeed, but nothing with the battery at LCE.
It's very difficult to accurately judge how much a camera repair will cost, and actually despite owning an alarming number of cameras I've never actually had an intermittent power problem before. Whatever you do, make sure you ask for an estimate to be given to you before any work takes place - OM1s are not hugely expensive, and you may have to weigh up the cost of service vs. replacement.
Thanks freecom and Nick. I'll check the battery terminal as well tonight. It's disappointing since the camera is in really good condition cosmetically.