The fuse is actually to protect the appliance flexible cord. It's time related characteristic is such that any fault that could cause the cord to fail will blow the cartridge fuse first.
bs1362 fuse time current characteristics
No fuse will protect a high impedance internal fault where the fault current could actually be less than the normal current. I wouldn't however expect a router to have any sort of signficant fire risk. Billions across the world are left on 24/7 (Anyone ever recall a report of one starting a fire ? ). Comparing one to say a tumbler dryer is frankly derisory. Tumble dryers catch fire because combustible fibres from the clothes get trapped in the exhaust system. Restricted air flow and inadequate air flow monitoring is a recipe for a fire. Miele tumbler dryers turn off if the air flow falls significantly.
Routers are generally connected via a power block using low voltage DC. In this case the output is near certain to be current limited to a value that will restrict the energy available to a level with insufficient energy to generate any significant heat in the block itself or the router. Bit like expecting a short circuited USB cable to generate a fire. It won't get slightly warm.
