Have a look at https://www.thinkbroadband.com/guides/how-to-stay-on... - there is mention of the BT Smart Hubs using a non-standard, or certainly difficult to source, connector size.
Nope you can get them, I had to buy one (well a pack of 25) years ago when one of our cats was having a bad day due to I had to move where they were sleeping when I was replacing some cables and they decided to chew stuff, sadly the BT Power cable was nibbled on so I lost about a foot of cable and plug.
Sure I could of just soldered the wires back together and used heat shrink tubing around the two wires.
But I just used my digital Calliper to measure the outer and inner dimensions and while they may not look the same they do fit perfectly like for example this one
Image of a rewireable 2.5 mm locking DC plug you just need to get the right size and they do exist.
I got an account with RS Components Ltd so I get all my stuff from there if I cannot get any quality components from Amazon due to RS are not exactly cheap, but the quality is there.
So saying that you could buy a 12V Power Bank and power the devices on that, so while power is on you are constantly trickle charging the power bank and when there is a cut in the power, the power bank will keep them powered.
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Paul