I have been quoted for labour only de-tile then retile bathroom, fit shower vanity unit with sink loo and mirror, fit six spotlights and new floor covering: £3500. We live in North Wales.
The quote you have received seems reasonable for the amount of work involved. With all jobs until you start taking tiles off walls you don't know whether the background is suitable to tile onto or whether further work will be required to provide suitable surface to fit new tiles too.
Also you do not say how big your bathroom is.
With bathrooms there are many variable and processes to be considered are pipes needing relocated or chasing into the walls
Post your job & get responses from local tradespeople. Read reviews before you decide to hire.
Seems a little high if you are supplying materials, but without knowing the size, shape and any potential problems I'd be hesitant to say its a ridiculously high price
Sensible cost for a good quality job, you can get lower but could be poor.
I will say it depends on size. It is a Normal price for London.