OK, alarm update. I know you're all gripped...
Battery definitely dead. It's leaked a little bit and the +ve terminal has eaten itself (clip on cable seems fine though). Date on battery shows it was installed just under 11 years ago (and it's probably been dead for 4 or 5 years already).
Battery (sealed lead acid, 12V, 7Ah) is £39.95 in Maplin or £14.50 (with free P&P) from eBay. Hmm, tricky choice, not! Ordered. I can take the old one to the special battery grave at the tip on Friday.
However, after opening the case I could stop the alarm by entering the user code (or "Master code") which I know. To shut the case I need the engineer code, which I don't know, otherwise the alarm goes off (and can't be stopped) because of the tamper switch being reseated and that's an engineer thing apparently.
Resetting both the master and engineer codes is easy though (manuals found on t'Interweb), just have to shut off the power by flicking the appropriate breaker (and before replacing the battery) and hold down two keys on the keypad for 2 seconds within 5 seconds of turning the power back on. That's easy enough to do (although I won't do it now as I've probably annoyed the neighbours enough over the last 24 hours) as I've got to wait for the new battery anyway.
Luckily it's away from prying fingers (of a 3 year old) so I can leave it with the cover wedged slightly open.