A spokesman for William Hill said:
"Following the tragic events in Camden on Monday evening, William Hill is extending full co-operation to the authorities as they seek to understand the circumstances leading to this deeply regrettable and unforeseen accident.
We would like to express our sincere condolences and sympathy to the family for their loss."
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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.