I'm in the final throes of bringing the house into the modern era and
in terms of electrics all that remains is updating our living room and
dining room sockets.

Use the current cable. Add as many sockets as you like to it. Keep it as a
radial. If it is 6mm cable, use a 32A breaker. If it is 4mm, you may need to
do some calculation to see if you can use a 32A or only a 20A breaker.
However, you're extremely unlikely to need more than 20A for two reception
rooms anyway. There's no need to upgrade to a ring. Things are different if
it powers more than the 2 rooms, especially if it involves kitchens.

Unless, of course, the old cable isn't PVC, but rubber or something else
awful (aluminium springs to mind).

Christian.