Indeed with 13 amp fuse and 240 volts the fuse should only be able to take 3120Watts before it melts.
BS1362 Plug Top Fuses will take considerably more than rated for quite a while before failing
A 13 amp fuse should sit happy as Larry with 3kw up it 
IIRC 4.5Kw for 15 minutes
there's a rupture / fail /trip curve graph for UK plug top fuses ,and others for MCBs etc
that doesn't take into account Chinesium made crap "fuses" that may not ever blow in the event of a fault 
We use a Ye Olde White night vented dryer to dry clothes ,hanging clothes on a line in the rain doesn't dry stuff much 
BUT , the vented steam/hot air from drying does go via a 4" pipe ,under the conservatory floor ,to the Koi pond building ,which heats that in winter .(aka heat recovery)
and a quid to have dry clothes to wear ,even in todays financial climate ,is OK by me 
25A for BS1362 13A fuses.........and should last a minimum of 10 seconds (and likely much longer!)
That said, more than 13A and the fuse will 'age' quite quickly (hence why 135A MIG sets occassionally blow the 13A fuse!)
Maximum UK continuous socket current is 13A and under the BS spec approved devices (And of course all devices sold/installed in the UK should be approved.....) have to operate at this for 8 hours without exceeding a maximum temperature rise. For me, a device which pulled much more than 8A indefinetly I would be avoiding and using a more suitable installation than a socket!
So in theory you could get (based on UK mains being 230V +10/-6% so 253V to 216V) 3.29kW to 2.8kW for a resistive element, this voltage variation gets exploited by some shower manufacturers who quote a 'peek' output rather than a typical to make it look more powerful!
Most tumble dryers made in the last 20 years are around the 2.4kW area, of course vented ones just run the heater pretty much constantly until the timer is up or the humidity in the exhaust has dropped to the turn off level