The resignations are being seen as a sign of bigger troubles ahead for May, who defended the deal before belligerent MPs in the House of Commons.
The UK prime minister stressed that a complete lockdown was the only answer to prevent the overrunning of the country's state-funded National Health Service, which would be a "medical and moral disaster" beyond the raw loss of life.
The challenge to May's leadership was triggered after 48 Conservative Members of Parliament submitted letters demanding a vote to the 1922 Committee, which represents rank-and-file Conservative MPs in the House of Commons.
If rebel MPs from his own Tory party and Opposition benches succeed in uniting and seeing the motion through, it would force Johnson's hand on seeking an extension to the Brexit deadline from the EU at least until January 31, 2020, in the event that no withdrawal agreement is agreed between both sides by the middle of October.
An alert government in Delhi should have begun government-to-government discussions with Moscow the moment it came to know that Russians were developing a vaccine. That is to say, almost an year's time has been lost, observes Ambassador M K Bhadrakumar.
May will now go head to head with Leadsom in an all-woman contest.
Britain is due to leave the European Union by October 31, with Johnson racing against time to strike a deal to meet that deadline.
The 62-year-old premier remains defiant despite the prospect of a no-confidence vote after Jacob Rees-Mogg, a leading Tory Brexiteer, piled on the pressure by submitting a letter of no-confidence to the chair of the Conservative's influential 1922 Committee on Thursday.
She stepped down after failing to win over her ministers with a revised strategy over her plans for the UK's withdrawal from the European Union.
Bishop Michael Nazir-Ali, who was Bishop of Rochester for 15 years, said the Queen took an oath to "uphold God's laws" when she was crowned.
The government of British Prime Minister Theresa May was plunged into turmoil.