In the UK, what is the difference between a stove, a range and a cooker?

At least two of these words have several different meanings related to cooking, but here is one set of definitions that would answer the question:

Stove (Brit) - "a hothouse for plants"
http://oxforddictionaries.com/definition/english/stove

Range - "a large stove with burners and one or more ovens, usually heated by solid fuel"
http://www.thefreedictionary.com/range

Cooker - (Brit) "any large sour apple used in cooking"
http://www.thefreedictionary.com/cooker

Essentially the three terms are interchangeable.

https://en.wikipedia.org/wiki/Kitchen_stove

I've never met that Oxford definition for stove. Stove and cooker can be interchanged for cooking on, but stove in that context is perhaps a bit old fashioned. Stove can also be anything for heating a place - but in terms of a thing that sits there and gets hot through burning something, but not central heating or anything electric. Cooker never has this meaning. A range is usually taken to be a very large built-in thing that can be cooked on, has at least one oven, and also supplies hot water. These are not very common now, and tend to burn wood or coal. Range can encompass the Aga, but those are usually called Agas. These are still in production, variously fuelled by "kerosene, diesel, biofuel, gas or electricity", and refurbished solid fuel ones are available. Like the old built-in range, the heat is permanently on, unlike cookers or stoves which are only turned on when activity is expected. The Aga and the range store heat and work on lower heat for longer time. The modern electric of gas cooker works on higher heat for shorter time. And doesn't supply hot water on tap, like the range did. RMG's ref, and https://en.wikipedia.org/wiki/AGA_cooker