Great news! At first test, the transplanted motor seems to work nicely. The sound issue was a nice easy one, the connections to the DIN plug were completely botched so I actually removed the DIN plug completely and just connected it up to RCA plugs for testing purposes. And it worked very nicely - although Søren is completely right, it needs a preamp both because it's quite quiet and also because it needs grounding to remove hum.
After reading through the manuals a bit more, turns out that the 1200 takes a preamp "card" that slots into the bottom of the unit. The thin grey cables that I thought were battery cables in fact connect to windings in the motor to produce 15V to power the preamp. I can't use that any more because the motor from the 1203 doesn't have that 15V winding. Not to worry, I'm sure an original preamp card for the 1200 wouldn't be that convenient to come across, so I'm just going to buy an external one. I decided to get a cheap one at first until everything was working nicely then I might invest in something a bit nicer.
Next steps (apart from waiting for the preamp to arrive) are to reconnect the switch and affix the mains terminal block to the chassis. I noticed a slight rubbing sound when the turntable is spinning (once per revolution) so would like to see if I can figure out what's causing that too. Then a final polish and hopefully we're in business!
Here's the mangled DIN connections: