Great shout !.
Do you have a three pin plug socket, on the same face plate as your electric cooker by any chance.
If so, try plugging the extension into that socket outlet, then into the "Granny" and try charging the car.
The cooker outlet is very likely to be on it's own dedicated circuit in the consumer unit ( fuse box ).
If this does NOT trip, then you have narrowed it down a little to a possible fault with the ring main circuits or the breaker.
If it still trips on the dedicated cooker circuit, it could be a earth fault that affects the whole house.
If your C.U. offers no protection, you will need to consider installing a new dedicated & protected circuit.
With a external IP65 rated wall socket outlet at the very least.