Surely if it has one socket direct and two fused it would be a tripler not a doubler.Old socket doublers often did have one socket direct and two sockets fused
I have only ever seen one doubler with a fuse (strangely it was the oldest one). Triplers always seem to have a fuse though I don't know if it covers all sockets or not.
Even with MK this is the case! their triplers are fused and thier doublers unfused.
I just checked the standard (being a uni student has it's perks one of which is easy access to a lot of british standards though unfortunately not BS7671)but to comply with BS 1363 all outlets must be fused
Regarding doublers BS1363 part 3 section 12.4 a clearly states
"An adaptor having only one or two adaptor socket-outlets for 13 A plugs complying only with BS 1363
and having no other adaptor socket-outlets need not be fused."
Triplers are covered by 12.4 c which is more vauge as it says a fuse is needed but doesn't specify what it should protect!
"c) A multiway adaptor having more than two adaptor socket-outlets for BS 1363 plugs shall be provided
with a 13 A fuse link complying with BS 1362:1973."
They don't seem to be as popular as they used to be but at no time have I tried and failed to obtain one. IMO the weight on the plug is only an issue if they are used stupidly (e.g. stacked or loaded with heavy wall warts).The weight of plugs fitted into socket doublers is the main problem and it can mean the plastic safety part of the pins gets compressed and allows the unit to push the spring grips upwards and lose connection with the lower part of the pin. I had though as a result these had been withdraw in favour of the multi-adaptor where a standard fused plug is connected by a short length of cable to a 2 to 10 way array of sockets.
I don't belive adaptors sensibly used present any real danger. IMO there is a lack of eduction though. How many people know how to find the rating plates on thier equipment and add them up? or make a realisitic estimate of the power use of a collection of equipment? How many even know equipment has rating plates?nearly every office one goes into has a multi-adaptor to run computer, printer, monitor etc. And with such a low current draw there is no real danger and if we feel multi-socket extensions are dangerous