Paul Lakin, leader of Rotherham Borough Council has resigned with immediate effect.
Cllr Lakin - who only took on the top role last year - will also stand down as local councillor for Rotherham's Valley Ward, the council said in a statement.
The rest of the cabinet are due to resign their positions "as soon as transitional arrangements are put in place", according to an earelier statement.
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Ms Wheeler and Mr Johnson married in 1993 and have four children. They separated in 2018.
HSBC employs 41,500 in the UK, but the bank has not said how many jobs will be cut in Britain.
There had been confusion amongst the government as to whether the Budget statement would still go ahead as planned.