Before you search for anyone, find the stopcock. In most UK homes it is under the kitchen sink, though in older houses it can be in a downstairs loo, a cupboard by the front door, or under the floorboards near where the mains pipe comes in. Turn it clockwise until it stops. If it is stiff, use a cloth and steady pressure rather than force, because a stopcock that has not moved in twenty years can shear. Once the mains is off, the water coming through the ceiling has a limit to it, and the evening changes from a flood to a mess.
A closed stopcock stops fresh water entering the house. It does not empty the pipes, the cold tank in the loft, or a hot water cylinder, and all of those keep draining down to the leak until they are done. Open the cold taps downstairs to pull water out of the system faster. If the leak is coming from a heating pipe or a radiator, the mains stopcock will not touch it, because the heating is a sealed loop; switch the boiler off and let it cool. The volume in a heating system is finite and the plumber will drain it properly.
Turn the electrics off at the consumer unit for any circuit the water is near. Put a bucket under the drip and, if the ceiling is bulging, poke a small hole in the lowest point with a screwdriver so it drains in one place rather than bringing the plaster down.
Be clear with yourself about what you are buying. A night call-out is a plumber coming to make the situation safe: find the leak, isolate it, cap or temporarily repair the pipe, and get you to a state where you can turn the water back on for the rest of the house. It is rarely a full repair. If a burst pipe needs a section replacing behind a tiled wall, that is a daytime job with the tiles off, and a good emergency plumber will say so rather than start chasing tiles at ten at night.
That also tells you what to ask on the phone. How soon can you be here, what is the call-out charge, does that include the first hour, and what happens after that. Firms that do this work properly answer all four without hesitation. Ask also whether they carry stock for common fittings; a van with push-fit and compression fittings, a few lengths of pipe and some isolating valves can fix most bursts on the spot, while a plumber with an empty van is a second visit waiting to happen.
Once the stopcock is off and the system has drained down, a lot of leaks can wait. Picture a pinhole in a pipe under the bath: with the mains off and the taps run dry, nothing is coming through by half past ten, and a plumber at eight in the morning costs a fraction of one at eleven at night. The exceptions are a leak you cannot stop (usually a fault on the mains side ahead of the stopcock), water near electrics you cannot isolate, or no water at all in a house with young children or someone unwell. Those are the ones worth paying the night rate for.
If it can wait, spend the evening usefully. Photograph the damage for the insurer, note where the water first appeared, and write down what you did (stopcock, taps, electrics). The plumber in the morning will get to the cause faster with that information.
There are 3,554 plumbers listed on this site, and the listings show which ones offer emergency and out-of-hours work as a service rather than as an afterthought. You can browse plumbers near you and phone the ones that state 24-hour cover, or describe the problem and request quotes for the daytime repair once the emergency is contained.
Tonight, in this order: stopcock, taps open, electrics off, bucket. Then the phone.