Unite joined the protest in London against the closure of the company which makes black cabs.
Unite's Peter Rose said: "We're here today because we want to campaign to keep the black cab built in Britain. It's been here for over 70 years and we want that to continue. It's a London icon, deserves its place in London."
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A lamp in the tower which houses Big Ben is to be switched off for the first time in more than 70 years.
Sir Bruce entertained audiences for more than seven decades.
Work in the tower will also pause over the bank holiday weekend.