Is it rude to talk on a mobile phone for the whole journey, when travelling alone in a London black taxi?

No it is not rude. I've lived in London most of my life and travelled in hundreds of black cabs and can assure you that cabbies really do not care what you get up to in the back of the cab -- as long as you pay the fare, give them a decent tip and don't spill food/drink/bodily fluid.

My intuition as a Londoner was that this was absolutely fine, but then I had a moment of self-doubt and worried that maybe I'd been being rude in not talking much to cabbies all these years.

So I sought out this ethnographic study of cab drivers:
Inside the Mind of a Cabbie (RSA)
and that has confirmed my impression that the time and space are basically yours to do what you like (within reason), and you don't have much in the way of conversational obligations towards the cabbie:

PASSENGERS AS PAYING GUESTS

Cabbies describe their experience of being in the cab as ‘working from home’ (see below), and in this context passengers are their ‘paying guests’.

Each passenger is a ‘gamble’, because they may be low or high paying and have good or bad behaviour. However, each passenger is accepted from the ‘taxi god’, and it’s part of drivers’ etiquette to accept the luck of the draw, although they reported on ‘other’ drivers who became upset by low-paying passengers/fares or local ‘bell jobs’ (low paying fares).

Overall, cabbies worry about the reputation of the trade, and they maintain or improve the reputation of taxi driving through their passenger interactions. This service side of the job includes intuiting whether the passenger wants to converse, treating them with politeness and respect e.g. keeping their cab clean, carrying luggage, not talking until they’re sure the passenger wants to converse, maintaining suitable boundaries (not discussing ‘deep’ topics relating to religion or emotions) and dealing with problem passengers in a responsible way.

Problem passengers may question the fare/route taken, escape without paying, be drunk (becoming aggressive or comatose in the back), or sometimes abusive for no reason.

Drivers intuit whether passengers want to chat by asking specific questions and gauging the response, or by responding to the passengers’ initiative:

“Most of them are nice, they’re like ‘how are you?’, most of them talk… but some of them have had a bad day and they don’t want to talk… if I see they don’t want to talk I leave them alone.”

(p.11)

So it seems like the best way to be nice to cabbies - the thing which they'll appreciate most - is to use their cabs for long, expensive journeys, and give them big tips. Chattiness appears to be fairly far down the list of what makes for a good passenger.

Once a cab driver told me if you sit behind him that means you don't want to talk. If you sit diagonaly in the back, you want some chating, and if you sit in front with him, you want to talk.