Anyway, he talked me through the mechanisms of the trike, and I actually know how some of it works now, which will be useful in maintenance. More importantly, he showed me how to fix my gears. It turns out that adjusting them is a fairly trivial process which requires no tools. We also changed the seat position a bit, to make it more comfortable - seems to be ok so far.
We went to New Malden, which involved a toucan crossing over the main road, a contraflow cycle lane, a semi-main road, a big hill, an extremely busy set of traffic lights (but it's outside the hospital!), three or four mini-roundabouts, a level crossing, and a narrow bridge. I think I'm now fairly well-equipped to handle most normal roads ;) Still a bit afraid of Kingston one-way system even with all the cycle contraflows, but I'll have another lesson in a few weeks to learn my way round that.
I do highly recommend cycling lessons to anyone who wants to learn to ride a bike safely on the roads. The instructors deal with people from complete beginners right up to experienced cyclists who want help with things like the Hanger Lane Gyratory system (put it this way - I know people who have been driving for 10 years who won't go near it in their car, and not just because of the inevitable tailbacks). No matter how bad you are, they've probably seen worse. If you have no confidence to ride on the road, they'll meet you in a park where you practise your balance without any traffic. They're not going to laugh at you for being crap and wobbling all over the place, because it's their job to help you. Also, if you book lessons through your local council's Road Safety department, it'll be heavily subsidised - maybe £12 for an hour of one-to-one tuition for an adult, or £20 for 6 hours of group lessons.