Ok, it's all fixed! It was clearly the relay that had failed, despite the garage telling me the motor needed replacing. As this thread is now showing up on a Google search for Auris wiper relay, I should probably just outline how to fix it here for anybody running into the same thing.
The part you need is a relay 85940 02040 and it looks like this:
Should be around £15 from a breaker, obviously a lot more new. Mine was from ebay and is a genuine Toyota part.
It took a while to work out where it was on a RHD car (as most schematics are for the LHD version), although it turns out the position is similar in both cases. It's behind the lower right panel below the steering wheel, where the mirror controls and headlight levelling dial is. This just pulls off very easily:
In this photo you can also just about see the double-locking door relay (which is blue) in case you ever need to get to that. Anyway, the windscreen wiper relay is dead centre once you have this panel off:
Really easy to access, just one bolt and out it comes. Swap the wires and that's it. One of the easiest repairs it's possible to do on a car like this.
Took about 5 minutes and cost £15. A bit better than paying a garage over £175 to muck about with the motor (or they'd probably just change the relay and then charge me the same anyway). The wipers now park correctly, auto rain sensor works, wash/wipe works - all good.
Thanks to everybody for their advice, and particularly the link to PartSouq which is where I started. For anybody having trouble locating the RHD relay locations guide, here it is: