Hi Kate and Welcome to A1
One of our more qualified Accountants will possibly correct me here, but I would of thought it is a 'swap' only in name, and that you have sold the vehicle therefore VAT will be due and purchased a private vehicle from a non VAT party.
Don't take that as gospel though. I am sure Peter, James, Boxby or similar, will be along soon with a definitive answer.
All my best
IP2