Presumably, a vehicle that, by virtue of it's weight, would have to be plated and tested as opposed to any other class of test, would be exempt from plating and testing if it is pre 1960, used unladen etc. etc.
I always assumed it needed to be a good vehicle to claim exemption but reading the form it states a motor vehicle, pre 1960 etc. I think this covers any large vehicle that would normally have to be plated and tested. So although a recovery vehicle is exempt anyway, I believe it still would be if it meets the weight/ age/ use criterion simply because if it were tested, it would have to be a plate and test and it is exempt from this requirement if used as above.
As class VII only covers up to 3500kg, the only test applicable is a full HGV test.