Odd, I've never experienced anything like that with the NHS, nor heard of our from anyone I know. There will be a full record of your treatment so this sounds like a very risky thing on the part of the practice.
I thought it might be that many practices are both NHS and private and you can have some things covered but get some private work done at the same time.
I can't comment on the taxis much but all will be regulated with the local council so that would be a point to start a complaint. My only note is that some cities like Liverpool or Manchester can have major congestion which needs avoiding, as well as (particularly on Liverpool) seemingly insane one way systems.
Uber and some other local versions send you a receipt with a map of the route taken which you could complain with.
Consumer protection laws are pretty strong here. I can dig out mode specifics if it'd help.