These vehicles, commonly called "black cabs" are still common on British roads. They are also called "Hackney cabs" or simply "Hackneys", but none of those are brand names.
There are two types of licensing for taxis in the UK, one for Hackney cabs and one for "radio cabs" or "radio taxis", sometimes called "private hire cars". The difference is that you can simply stop ("flag down" or "hail") a Hackney cab if you see one passing and they can stop and pick you up. They have a light with the word "TAXI" on the roof, and if the light is on, they are free to pick someone up. If they are already carrying a passenger they turn the light off. Almost all Hackney cabs are of the kind in the picture, although they are not all black. Radio taxis, on the other hand, are not permitted to simply pick up passengers on the road. You have to phone through to the office, and the company will send a taxi out for you.
The word "Hackney cab" comes from the older, horse-drawn "Hackney carriage".
As far as I can see the one in the picture is a TX4, built by the London Taxi Company. The original design goes back to several successive designs by Austin, starting with the Austin FX3, which dates back to before WWII. The single most iconic model of black cab was the Austin FX4, which was built by different manufacturers from 1958 until 1997.
You will find plenty of information on the various Wikipedia pages: