I'm in London and I'm not paying anything like that much for my bathroom!
We've got a labour only quote for £3.2k (three quotes, cheapest was £3.2k, most expensive was £4k), then you're looking at fittings and tiles. As you say, the sky can be the limit but it's really not necessary to go for Fired Earth tiles and Perrin and Rowe fittings; lots of DIY stores do perfectly serviceable and inoffensive (and even quite interesting) tiles and fittings for far, far less money.
It looks cosmetic, and you're more than capable (disabilities aside) of painting and decorating yourself when you're in. Get boiler information and an electrical test carried out, make sure you have a full structural survey done. Those things will drive your essentials and you can get quotes for anything that comes up.
If you want to do the bathroom and kitchen before you move in (or early on in living there), budget maybe £10k for the kitchen (not high end but perfectly nice, have a look at DIY kitchens) and £5k for the bathroom.