Labour divisions have burst into the open once again as two shadow ministers hinted that they would resign if Jeremy Corbyn succeeded in dropping Labour’s support for renewing the Trident nuclear weapons programme.

Owen Smith, the shadow work and pensions secretary, said it would be difficult for him to remain in the shadow cabinet if the party supported a policy of unilateral nuclear disarmament. Lord Falconer of Thoroton, the shadow justice secretary, said he was a strong supporter of Trident.

And in a further sign that the leadership is struggling to impose its authority after the reshuffle, a third member of the shadow cabinet dismissed claims by John McDonnell, the shadow chancellor, that nuclear policy could be changed through a new process.

The interventions will come as a blow to Jeremy Corbyn who had hoped that last week’s lengthy reshuffle would lead to greater coherence in the shadow cabinet on foreign and defence policy.

Jeremy Corbyn defends reshuffle with call for Labour unity Read more

Hilary Benn kept his post as shadow foreign secretary on the understanding that he would consult more closely with Corbyn. Maria Eagle, a strong supporter of Trident, was removed from the post of shadow defence secretary to replace Michael Dugher, who was sacked as shadow culture secretary. Eagle was replaced by Emily Thornberry, who opposes the renewal of Trident.

McDonnell, who is Corbyn’s closest ally in the shadow cabinet, made clear during the reshuffle that a move towards embracing a policy of unilateral nuclear disarmament lay at the heart of the reshuffle. In a Channel 4 interview last week, he said Labour’s national executive committee was examining plans for the policy on Trident to be decided by a new “consultative process”. This would involving bypassing the Labour party conference by holding a ballot of party members. This would allow for a change of policy ahead of a parliamentary vote on the renewal of Trident, which could be held before the summer.

In a direct challenge to the Labour leadership, Falconer and the shadow education secretary, Lucy Powell, dismissed the claims by the shadow chancellor. Falconer said the Labour party conference, where Trident-supporting trade unions have a key say, decided policy.

“I don’t know where the briefing is coming from,” Falconer told the BBC’s Andrew Marr Show regarding McDonnell’s intervention. “The way that policy’s made in the Labour party is ultimately by conference. For example, in relation to Trident, there are very, very, very many people who are employed in industries that support the Trident defence system. The unions should have a role in determining what our policy in relation to that is and I’m not aware of any detailed proposals that we change the method by which we make policy.”

Powell echoed Falconer’s remarks as she dismissed McDonnell’s claim about a possible change to deciding on policy. She told the Sunday Politics show on BBC1: “I don’t think that’s going to happen and I don’t believe those briefings for a minute either.

Facebook Twitter Pinterest Lord Falconer, the shadow justice secretary, on the Andrew Marr Show. He criticised the sacking of the ‘absolutely exceptional’ Pat McFadden. Photograph: Jeff Overs/BBC/PA

“Any changes to Labour party policymaking process, as those on the left will know better than anybody because they’re usually the holders of the rule book if you like, will know that changes like that can only be made at Conference by changing the rule book of the Labour party,” she said. “We have a very consensual policymaking process.”

Owen Smith indicated that he might have to leave the frontbench when he was asked on Pienaar’s Politics on BBC Radio 5 Live if a change in policy would lead to his resignation. The shadow work and pensions secretary said: “Well, that would be difficult for me but I think the key thing that I would do is stick in, in the run-up to that decision, and make the case.”

The semi-rebellious mood among some shadow ministers was highlighted when Falconer rebuked Corbyn for sacking the “absolutely exceptional” Pat McFadden from the shadow cabinet. Falconer said he wholeheartedly agreed with McFadden’s warning last year about drawing a link between terrorist attacks and western military intervention. McFadden was told that his remarks, in which he spoke of the dangers of infantilising terrorists, had been seen as disloyal to Corbyn, who spoke after the Paris attacks of the legacy of 14 years of western military intervention in Iraq and Afghanistan.

Alison McGovern, a former parliamentary aide to Gordon Brown, resigned from a new party policy review examining child poverty, after McDonnell attacked a “right-wing clique” of Labour MPs associated with the Progress group. McGovern, the Progress chair, said McDonnell was late for a meeting to discuss the work of the review last week because he was recording broadcast interviews criticising her group.

A Labour source described McGovern’s move as an “abstract” resignation because discussions about the child poverty review were at an early stage. The source said the leadership hoped McGovern would return.