What is the British nickname for toilet?

It's called a "toilet". It should be noted that in the UK, "toilet" can also mean "a room with a toilet in it, or several toilets" as well as the actual device. In the UK, a "bathroom" is a room with a bath inside it.

There are loads of different Words, both formal ("WC", "Lavatory") and informal ("Loo", "Bog", "Men's/Ladies' Room")