Tory MP Nadhim Zahawi has written to the chief of the Met Police, Sir Bernard Hogan-Howe to ask him to investigate the comments by Liberal Democrat MP David Ward.
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Commuters were also warned to expect travel chaos as Doris gathered speeds of nearly 90mph.
A man with 62 penalty points on his licence is among thousands of motorists allowed to drive despite exceeding the limit warranting a ban.
Storm Doris is set to bring severe weather across much of the UK on Thursday as heavy rain and strong winds push in from the west.