First, is it a side by side or one with the fridge over or under the freezer section?
In any case, try this... when you hear the unit turn itself on, open the door and determine if you hear a fan running and feel cold air being moved about both the freezer and fridge.
This question has been asked numerous times and my response is based on experience here in the U.S. Most people don't realize that any standard fridge/freezer combination uses cold air only supplied to the freezer section. A thermostatically controlled valve and door allows the desired amount of cold air into the fridge side. Often it's this valve controlled door that fails.
If it is the thermostat, it'll be directly under the round dial that you probably turn right or left to increase/decrease the temperature. The knob can usually be pulled off and this will reveal a method to access the 'stat. It's usually held in by three or four screws and is supplied with electric wiring, so be sure. to unplug the unit before working on this and be sure to mark which wire goes where (they're differently colored)....
Here in the U.S. a new thermostat will usually cost around $30 to $50, I wouldn't expect them to be anymore expensive in the U.K.
You'll need the part number off the old one or better yet, after removing it take it to a supply store for replacement.
But again, I'd emphasize testing the unit to see if any cold air is coming into the freezer side before doing anything else...
Good luck!