My boiler has died and to get a new one is going
The landlord has to repair in a reasonable time frame. The law isn’t specific on what is reasonable but I would say 1-2 weeks is reasonable. It isn’t easy to get tradespeople quickly, I’m a landlord but my boiler broke recently under warranty and british gas took 2 weeks to fix and there is a shortage of available tradespeople.
Luckily it’s not january but appreciate its far from ideal. You could ask your landlord ror a reduction as a goodwill gesture and/or supply temporary electric heaters. Landlord doesn’t have to do this but I would for a good tenant.
You should continue paying rent and at this time of year you are highly unlikely to be put in a hotel.