Unlike a lot of other heavy appliences tumble dryers will draw close to their max rating for most of the cycle (this could be 80 or 90 mins)
So, if that dryer's rated 2050 W you can assume that it is actually consuming that much power for almost all of the cycle.
I just had a look at our Miele dryer and it's also a 1.5 X 3 flex and it's slightly warm, but not hot just noticably warm.
The plug is also slightly warm, but not dangerously so.
Your washing machine will only draw the max rating during the heating part of the cycle which might only last 15 mins or so. The element will be intermittently turned on and off for short times to maintain that temp during the wash.
However, there is no way that you can run both of those appliences together on a single 13A plug! The fuse *SHOULD* blow.. I'm amazed it's not blowing.
Running both of those appliences on an extension cord from one socket is a potential fire hazard.
2050W dryer + about 1850W for a washer is considerably more than 13A
I've seen a BS1363 socket catch fire and it's not something you want to have happen!
Regarding the pins heating up:
British plugs and sockets are not recessed, so to prevent electric shock most of the surface area of the pin is sheathed in plastic and the sockets are designed only to make contact with the tip of the pins. The surface area of the pin that actually makes contact is supprisingly small given the size of the plug.
(European schuko plugs don't really have this problem as they're recessed)
Anyway, for heavy appliences like tumble dryers I think it's very important to ensure that the plug and socket are of good quality and are making proper contact. Repeated plugging in and plugging out will eventually loosen the contacts in the socket. Personally, I would always make sure that the outlet for dryers, washers etc is a good brand like MK (likewise for the plug).
I honestly don't think that BS1363 plugs are a very good design. They're far too prone to overheating and rely on far too many add-on safety devices (sheathed pins, fuses etc). A modern recessed schuko plug / socket system is just fundementally safer.
There are a number of things that could have caused that 18 year old dryer to burn you / over heat the plug.
1) Older dryers didn't have very good temp controls so tended to draw more power as they just kept the heaters on constantly throughout the cycle. Modern ones would tend to switch in out thermostatically.
2) The plug or socket could have been making poor contact.
3) Given that it was 18 years old, it could have been designed for the rest of Europe (where sockets / plugs can provide 16A) or with the older BS546 15A round pin plugs. So could easily have been drawing close to 13A all the time.
Any of the above, or a combination of the above could have caused the problem.
When BS1363 is used in Ireland, on 220-230V systems this problem can be even more pronounced. There are plenty of instances where 15A BS546 plugs and sockets, schuko in older installations, or direct hardwired connection into a 16A circuit is used.
