Is the “plumber” a qualified plumber or a drain clearance person? You need to know why the loo and shower blocked not just what blocked them. Has he checked the plumbing for a cause or just told you what he cleared out?
Unless the shower plumbing was already blocked when you moved in (did it drain very slowly?) or the plumbing has some issues that mean that even a few stray hairs getting through the hair trap (or there is no hair trap) are enough to cause a blockage, it is likely it is your fault for not regularly cleaning out the hair.
Poo alone really shouldn’t block a toilet. Either you or another tenant must have flushed something down the loo that caused a partial blockage (wipes, pads, food etc) or there is an issue with the system (a bend in the pipes, too narrow pipes, inadequate flush). If it’s the system, it’s the landlord’s issue.
Given that both failed at the same time, it might be the soil pipe they drain into that is blocked (which could be shared with other properties. Or the plumbing is dodgy.
If you don’t have another toilet, the landlord needs to sort it or provide an alternative within 24 hours because it is a health risk.
The landlord has to provide an adequate heating source for each room. You really need to get legal advice from Shelter or CAB or your local council if your home has serious mould issues and inadequate heating. Is everything else in order eg gas and electricity safety certificates, deposit protected, EPC, right to rent documentation etc?
TBH if the landlord isn’t bothered about legal requirements for heating, I wouldn’t be surprised if the plumbing has major issues too.