Following the discussion about Schengen, 90/180 days, etc, I asked a friend who travels abroad for work with Rolls Royce if he knew anything useful.
He had no knowledge about visas, but said:
"Our travel office are advising that despite Brexit there are no visa requirements for any European states. However, some are now insisting that those driving into the country will need IDPs (Get one at your local Post Office ... your driving licence, passport photo and £5.50 required). If you are flying into eg. Italy and hiring a car there is no need for IDP. You should check this before you set off ... or just get an IDP in case."
I thought that I wouldn't need an IDP for taking the 'van to Europe, but it seems that I might have to get one (or two if I want to go to Spain *and* Portugal)!