Hello young lady/man,
Sorry to hear about this problem - unfortunately, today in UK [probably in the world??], every well-off person/company is a crook - unfortunate as it is, when dealing with these crooks, there is no alternative but to be crooks vs them ourselves.
However, first try persuasion by telling this landlord that his water bill and heating bills will be much reduced if he/she has a shower fitted for a lot of people [5 of u].
If that does not work, write to the local council [cc to landlord] who are responsible for ensuring that landlords behave and provide habitable places for their tenants. I know it sounds unethical, but in this letter say that one person sharing your flat has a major problem with his/her legs e.g. that there is limited flexion at the knees, and cannot get into the bath, so a clean, safe, working shower is crucial.
I am not a young man so I have seen the transformation of businesses in Britain into fraudulent deceivers [you will remember Tesco as well as NatWest were caught red-handed cheating customers a few years ago - there was coverage on TV], so I have no time for such blatant extortionists - I would urge you to be tough!
Good luck!
M