In a Transport Select Committee meeting held this afternoon, Willis said a quarter of the 1.2 million cars that were running with software deemed to cheat emissions tests had already received new software.
Willis said he hoped the remaining 730,000 affected cars in Britain would be fixed by this autumn.
The meeting, which featured representatives from the Department for Transport (DfT), revealed that 3,500 customers who have received the new software had complained about negative changes to their cars. Complaints related to the way the car drove or the noise it makes and reductions in fuel economy.
Willis said that Volkswagen has paid £1.1m to the Department for Transport to cover the costs of retesting following the emissions scandal.
The rising costs from fines and legal battles facing Volkswagen from its emissions scandal will lead to the cutting of 30,000 jobs. The company has been hardest hit in the US, where it will pay at least $1.26 billion (about £1bn) in fines and have to fix or buy back almost 80,000 cars.