Advice from appliance manufacturers is that if the appliance is under 700W, it should have a 3A fuse fitted, if over 700W, it should have a 13A fuse fitted
That advice - whilst better than using a 13A fuse for everything - is IMHO, badly worded, technically inaccurate and does not consider the bigger picture.
The majority of our electrical items are covered by 3 or 5 amp fuses.
Only the heavier load appliances have 13 Amp fuses installed.

Absolutely.
My 2p worth... if anyone is awake...
A fuse is required to protect circuits from overcurrent situations and the subsequent risk of fires from the overheating of wires and components in the circuit.
For any given device, the power it draws should be known or determined Based on that power draw the size of the cable supplying the device should be calculated, allowing some leeway.
The power that a given cable can safely handle will be a result of its material, cross-section, max temperature characteristics of its insulation and the method of installation (e.g. whether it is in conduit, under insulation or in open air).
The fuse used on the circuit supplying that cable should therefore be rated less than the maximum carrying capacity of the cable, to protect the cable from overheating in failure conditions.
My 'rule-of-thumb' is that for a given current draw of a device, then the fuse should be rated at that current + 20% and the wire supplying the device should handle the fuse current + 20%.
So, taking the example of a 900W toaster. The current draw at nominal 230V will be 3.9A. Adding 20% gives a fuse size of 4.7A. Therefore a fuse rating of 5A is appropriate. The wire used to supply the toaster should be (assuming regular domestic 3 core flex in open air) a minimum of 4.7A + 20% which is 5.6A. Therefore flex of size 0.75mm² would be suitable as that will typically handle 6A.
But if, for example, the 900W toaster you buy comes with 1mm² flex, that cable will have a current carrying capacity of around 10A. Which is all good and generously sized for the toaster's power draw. BUT 1mm² cable should not be used with a fuse of 13A. The fuse should blow before the cable melts should the end device fail and draw too much power. Hence a fuse of 5A or 10A should be used, not 13A as that linked advice suggested.