The fuse in a BS 1363 plug is BS 1362, and they come as 1, 2, 3, 5, 7, 10 and 13 amp, for a fixed appliance the manufacturer can stipulate fuse size and type, but when we joined the common market, portable equipment had to be made so it could be used in other member countries, and other than the UK it is rare to have a fused plug, so if portable equipment needed a fuse, it has to be inside the equipment, so the fuse in the plug only protects the cable, so we reduced the common BS 1362 fuses to just 3 and 13 amp in the main, as they protected most of the common flex cables used, but this is not extended to fixed equipment, many boilers stipulate a fused connection unit with a 3 amp fuse. (Personally, I prefer a plug and socket, this allows the heating to be powered from an inverter or generator in a power cut)
Before we joined the EU, I tried to show my children how to select the correct fuse, and it is not as easy as it first seems, so 700 watt would with a resistive load need a 3 amp fuse, but with a motor, it could need a 5 amp, the fridge and freezer are the prime examples, the rating plate may call for a 1 amp fuse, but the start amps would rupture that, so more likely it needs a 5 amp.
Try explaining this to children, and then have them return home from school, and tell you their teacher says I got it wrong. This was before everything had to come with a 13 amp plug, so school children were taught how to fit a plug. I have had this a few times with the kids at school, the classic was there are two types of transistor can anyone in the class tell me what they are, my son, who had passes his RAE answered bipolar and field effect sir. To be told no, it is NPN and PNP, to which he burst out laughing, he did not even attempt to tell the teacher his error, and was still laughing when he told me.
Just happened it was parents - teacher night, so the teacher was told, he said "Oh field effect must be new?" to which I told him no, invented before bipolar, however missed out it was not perfected until latter.
I did google "how to select correct fuse for a uk plug" and the answer was "To select the correct fuse for a UK plug, check the appliance's power rating: if it's under 700 watts, use a 3 amp fuse; if it's over 700 watts, use a 13 amp fuse; always consult the appliance manual for the most accurate fuse size." not so different to what I was telling my children all those years ago.