The socket seems to resemble the 3pin 15amp (and 5amp) round in sockets used pre 13amp plugs... Is one pin larger than the other two - which it ought to be - as that pin will be the earth/ground connection for the load and also acts as a locater (ensuring that live and neutral are connected correctly to the load...). In the days of round pin plugs it was often possible to have two pin plug which fitted into a 3 pin socket and have the live and neurtal inadvertently switched... On a basic isolating transformer - 240 in 240 out - not a step down transformer - there may not always be an actual earth for the load...; volts out being truly isolated from the input side but no earth for the load via the larger third pin. Other versions did/do have an actual earth connection via the third pin and it gets that from the input side of the transformer. Much as today most, if not all, site transformers have an isolated o/p and the third pin is an earth connection via the input.
Bearing in mind the apparent age of the transformer I'm inclined to feel it's a standard 3pin 5amp socket requiring (the then) standard 5amp three pin plug; preferably rubber or similar.
So what you likely have is a 240volts in 110volts out (step down) isolating transformer - hopefully with an earth...
Today such transformers have shielded/skirted sockets and plugs of course with a locate in the skirt/shield, but still with that third larger pin..
You might try online for a 15amp or 5amp - whichever size (I suspect it's the 5amp) - three pin plug and see if it will fit; something like this:
https://www.amazon.co.uk/MASTERPLUG-Black-Round-Mains-Plug/dp/B07MK4M2WW
https://www.amazon.co.uk/Round-Plugs-Mains-Lighting-Stage/dp/B08CRTPYQH?th=1
As for replacing the existing socket... on something of that apparent age I'd be cautious. It may not as easy as you think... and if you get it wrong...
- is an excellent explanation of isolating transformers; which is what you have. View it to the end as JW covers all the variants.