Success!....almost. Thanks to everyone for the advice.
I stripped the wiper mechanism. The motor was running fine, the shaft was free turning and the cog seemed in good shape; however, there was almost no grease in there and rust and crud were increasing friction enough to prevent it working. Disassembly, a good clean and lube with ACF50 and it worked well. I packed it with grease (including the shaft) and put it all back together. The plastic back was held together with cable ties as the clips were broken, so I sealed it with silicone sealant as a backup.
Unfortunately as soon as I tried to run it with the wiper arm attached it became clear that the splines in the arm were toast. Worse, the additional load seemed to cause the original problem to reappear. The wiper mechanism may have too much wear to deliver the torque required. The shaft on which the cog turns is kept tight with a press fit circlip and I may not have gotten it tight enough.
I'll start by replacing the arm as that is definitely trashed, but this will of course require stripping the rear door again to connect the washer fluid line. I suspect I will end up having to replace the motor as well.
