This is easy.
As part of the Landlord and Tennancy act 1985 section 11 the landlord has certain obligations. Section B will help you but is more to do with safety but part C you can win with as it is seen as essential to provide space and water heating.
(b)to keep in repair and proper working order the installations in the dwelling-house for the supply of water, gas and electricity and for sanitation (including basins, sinks, baths and sanitary conveniences, but not other fixtures, fittings and appliances for making use of the supply of water, gas or electricity), and
(c)to keep in repair and proper working order the installations in the dwelling-house for space heating and heating water.
Do not withhold rent but do report it to the landlord again who is allowed a ‘reasonable’ time to fix the problem (baby and winter may make this a short time). Then seek to get money back via rent abaitment.
Also if you rented the place with hot running water that is what you expect to have, it will have been part of you decision to accept the place. No one would rent a place with no hot running water.
Mate or not I would be making a large issue for him.