My Fluke 87 III is now fixed and working!
The fault was "continuous beeping" which turned out to be the "you've left probes plugged into the current socket and you're now in V/Ohm" alarm. The Fluke uses split conductor input receptacles the sense the banana plug is inserted and this was faulty.
There are a few other little issues - the display clips are broken on one side but once the meter is back together, the pressure keeps the display working a treat and the Capacitance range seems a bit off - but everything else checks out in comparison to my Flukes from work.
Just a big thanks to VUK for his assistance in getting parts and for Celemetrix for being local and able to supply them at a better price than anywhere. Also thanks to ModemHead for his interesting and useful website.