Its a Beko fridge freezer, only 4 years old (should I expect better from appliances these days?) and the freezer compartment has not been cooling properly for a couple of weeks.

It doesn’t appear to be doing anything now, although the fridge light still comes on.

Is this a fixable problem or do I need to go shopping tomorrow (and eat lots of pizza and ice cream tonight?)

Where you keeping it? Had the same prob from my freezer when it was in the garage, turned out it’s thermostat was keeping temperature relative to outside temperature.

Took it inside during Winter, it was fine again.

Its in the kitchen and has been since new.

TBH it could be any number of things.
Hold the door open and see if the compressor starts (black “pot” at the back) You should feel it vibrate and hum. If it starts and does not cool then it could have lost its gas (check that one of the pipes is cold). If it doesn’t start then its probably the stat or the thermal overload. Thermal overload (klixon) U/S = FUBAR. Stat U/S = repairable.
Please note that these are “probables” and it could be something different.