I recently changed the hot and cold valves for a sink in an upstairs bathroom. The taps are mixer taps. All of the water for the bathroom (and indeed house) runs off the mains. Hot water is provided via a combination boiler.

The water pressure in the downstairs has not changed and is very strong. However, there is no very little pressure upstairs from the cold water. If I turn the sink taps on, followed by the bath taps, the sink taps stop.

There has been no impact on the hot water in any part of the house.

I think that there must be some sort of air lock upstairs. However I have no idea of how to eliminate it. I have turned on all of the taps upstairs, including the simultaneous flushing of the toilet. I have left all of the taps open for a considerable period. Nothing seems to work. I would assume (but we all know what that can do) that the mains pressure would force the air lock out. That’s if it is an airlock.

What started as a friendly gesture for an elderly neighbour, has become a nightmare.

Any help or suggestions would be greatly appreciated.

Ain't it da truth? That nice little old lady on a fixed income that you want to help out on the cheap turns out to be a project of major proportions that's going to cost someone big bucks.
You haven't got a air lock, my friend. Pressure systems don't "air lock". To exhaust air from a pressurized system simply open up the top most faucet and bleed the air out. About the only time a system air locks is if your attempting to drain the system and don't open up a upper level faucet to relieve the pressure. I hope I've put the "air lock" myth to bed.
Now, on to your problem.
You didn' furnish any details, but I'm guessing this is a older home with galvanized piping.
I don't know all the steps you did to accomplish this but it's my guess that you disturbed some minerals built up in the pipe walls and it's now clogging the mains cutting down on the volume. That's about the only explanation for this,
"However, there is no very little pressure upstairs from the cold water. If I turn the sink taps on, followed by the bath taps, the sink taps stop."
If that's the case you're in deep doo-doo. The pipes and affected faucets will have to be flushed out and if that don't help then a repipe job isn't out of the question. I've been where you are and I feel for you. How's that saying go? "No good deed goes unpunished". Good luck, Tom