Some of the old fuse boxes had a special first fuse with reinforced contacts

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without that the limit is 32 amp = 7.3 kW which would not be very good for a shower, the main switch is often rated at 60 amp,

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so with an electric shower easy to rupture the main DNO fuse, you can have RCD protection as seen here

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but it is rare with that fuse box.
The EICR (electrical installation condition report) is the personal opinion of the tester, BS 7671 is not retrospective, although other regulations can be, so if fitted in 1980 and it was OK then, and nothing has changed, then still OK now, however unlikely nothing has changed, we tend not to use tungsten lamps today.
But anything new must comply with the rules today, so a new shower will likely say in the manufacturers instructions should have a RCD fitted, so really no option.
The landlord is responsible for the electrics, so he must authorise any work, and since this is heath and safety it must be in writing, even if on a fag packet, phone call no good, although text message is allowed.
A consumer unit is the new name given to a type tested distribution board, so would replace the fuse box, when the fuse box was fitted the type testing had not started, so it is not a consumer unit, but would be replaced with a consumer unit.
Today a consumer unit (CU) needs type A RCD with a 30 mA tripping current in 40 mS, it will also likely have a surge protection device (SPD) claimed to stop spikes from damaging LED lights etc, and may has an AFDD (arc fault detection device) which also tends to trip with over voltage so to some extent protects from loss of PEN, there seems to be some debate on exactly what is required, however for you the user the big thing is if only two RCD's are fitted then far more likely to trip to if all RCBO's are fitted (RCBO is MCB and RCD combined) and if the RCD trips then you could loose contents of freezer if not reset fast enough.
It will likely cost £100 more to have all RCBO, and nothing forces landlord to fit RBCO's rather than a pair of RCD's. Also unless some thing else goes wrong, it is unlikely there will be a safety issue with no RCD protection, I would not expect you to be drilling walls and hit cables. I have all RCBO's fitted, but I would pay for lost food if freezer failed, so for me worth while.
I have also seen here again and again in resent months how landlords are selling up, as the new laws are stacked against them, and they are saying enough is enough, houses are not the investment they were. And the residential caravan sites are filling up as the rules don't apply to caravans.
So yes it is a problem if a tenant rocks the boat, and some times better to turn a blind eye and have a house than point out what must be done and end up in a caravan.