A Type-B 32A CPD needs 5*In or 170A for instant trip, whereas a BS1362
13A needs around 7*In or 91A for "pretty quick trip". Quality & most likely age is a factor - a fuse dissipates about 8W at 13A. You'll be delighted to know dodgy BS1361 fuses exist: no sand, they don't 'arf go bang.
An MK double socket is rated to 19.5A continuously. It is indeed worth running a new spur from wherever to at least another double socket.
There is a small chance the "32A ring" might actually be 32A radial in
4mm. For some houses it can be much simpler than forming a ring in
2.5mm, so do not be surprised. There is a length limit because 4mm CPC in just 1.5mm (lawd knows why, could easily have made it 2.5mm but there we go). Almost makes you want to use 6mm, but getting twin of that into a double socket in 35mm backbox just isn't going to be funny.
If you want to use one-multiple 3-3.3kW appliances on an extension lead, use H07RNF in 1.5mm. That has 16A continuous rating, is pretty tough and the 1.5mm vs 1.25mm offers a bit of leeway re local heating. Some dryers have heater covers that get scalding hot on the back if the cable is nearby.
Moulded plugs on some appliances are just junk these days. I have a suspicion some makers are using different insulation (90oC rated) or assuming short duty cycle (kettle) to get away with ever thinner appliance leads. The result is whilst cable heating is obvious the plugs can run almost hot which doesn't help backbox wiring. My AEG runs about 3.2kW and after 2hrs the 1.25mm flex is like spaghetti, the plug roasting. No doubt the fuse tabs aren't great - put a proper plug on it and the temperature drop is very considerable. Putting two such plugs in an extension would probably get quite nasty - some cheap extensions have very thin busbar inside.
We should have done a miniature-body hybrid of BS4343 (round-pin) 16A radials by now for high current appliances, and likewise cookers which I can't help but think would benefit from a 32A plug-n-go. No doubt the AD BR P Stakeholder Crowd will buy a shedload of clipsal series 56 and create some statistics in order to mandate their usage.