I own a flat in a block of four - all the communal lights (PIR operated) run off the flats' supplies, as does the door entry system.
It sounds rather like a fuse/circuit breaker is tripped and that's removed power from the lights. That's made the emergency lights come on until their batteries ran out. In theory the batteries should just recharge once power is restored - but when deeply discharged, it's common for them to fail.
A few years ago, one of the flats changed ands and the new owner was having some work done. Driving past (I used to pass on the way out from/back in to home) I saw my tenant taking a screwdriver to the outside panel. The switch-fuse for the other downstairs flat was "locked off" with some red sticky tape
because we believe he was having the CU changed. One of the residents asked "can't we just turn it back on ?"
I politely pointed out that we had no idea what exposed connections there might be inside the flat, so no we couldn't.
So you need to find the supply (which could be anywhere, and in any of the flats) and reset the fuse/circuit breaker. Chances are that that's all that's needed.
As to it being the management company's problem, bear in mind that all the mgmt co will do is get someone in - and add their markup before billing the residents. Allegedly the contractors they use will often artificially inflate the bills and the mgmt co won't care as they are simply passing on the extra cost.
In our flats, while the freeholder is in theory responsible for things - we normally just get together and do it between the four of us (or 9 if it's something common to the whole development) and split the bill. That way we get to choose who does what and how much we'll be paying