I don't want to say "that should be easy to do," but I'd be shell-shocked if any company with experience doing navigation systems couldn't figure this out without too much trouble.

Also, even if GPS signals are hard to get in London, navigation in a fixed city can be calculated in lots of different ways. Allowing for some regulatory limits, a company (or an experienced third-party) could scatter their own transmitters around the city as needed, figuring out which frequencies give the best accuracy.

It should be easy to minimize turns across traffic and avoid traffic lights... but they seem to have plenty of trouble doing that.

Google maps already integrates with Uber. If they managed to get such connection, I'm not sure why a bigger black cab company operating in large cities couldn't get an agreement to include taxi/bus lanes on a separate app. Can't imagine Uber forced Google to cooperate with them...

Google Ventures are an Uber investor, I'm not sure if that's what sparked the integration but it probably didn't hurt!

I'm not sure if there's actually a comprehensive source for bus lane data - if there is, it would be really useful for motorbike users too (who can generally use bus lanes).

Isn't Google -- through Google Ventures -- a significant investor in Uber? That kind of may give them a different set of considerations than "a bigger black cab company" would have.

Sure and I think we're in agreement - Google proved it will do something nice for a taxi company if they profit from it. It's just a very specific kind of profit in this case.

Well, that's a simple software problem with street definitions. Much simpler to fix than what amounts to implementing a GPS/pathfinding system inside someone's head for even one driver.