© David Moir / Reuters
Leading figures in both parties are set to use the incident, which occurred in June 2016 just weeks before a House of Commons vote on renewal, to attack the government. News of the missile malfunction only emerged on Sunday.
The SNP is committed to opposing it on the basis of safety and security, as the nuclear submarine fleet is based in Scotland.
The Labour leadership is opposed to nuclear weapons, but the majority of its parliamentary party is in favor of renewal.
SNP defense spokesman Brendan O'Hara told the BBC there are political and operational issues which must be addressed, but warned "this is not a national security issue."
"The government can't, as they love to do, hide behind the national security smokescreen. The public, who are paying over two hundred thousand million pounds [US$249 billion] for this renewal, have a right to know if it works or not," O'Hara said.
Labour's Shadow Defense Secretary Nia Griffiths said a full explanation is due, while Shadow Chancellor John McDonnell said it is "extremely worrying" that parliament had not been informed of the incident.
Likewise former Labour Defense Minister Kevan Jones told Labour List, "If there are problems, they should not have been covered up in this ham-fisted way. Ministers should come clean if there are problems and there should an urgent inquiry into what happened."
In a car-crash interview on Sunday with the BBC, May refused to disclose whether she knew about the incident ahead of the vote on Trident. MPs ruled in favor of renewal by 472 votes to 117.
Instead she opted to say she had complete faith in Trident and that she thought "we should defend our country," with repeated references to Labour leader Jeremy Corbyn's opposition to Trident.
Senior military figures have also weighed in, with former head of the Royal Navy Lord West of Spithead writing in the Daily Mail that this had been a cover-up "worthy of North Korea."
"The decision to withhold news last summer that a Trident missile test experienced some kind of problem - ironically, almost certainly minor - is both bizarre and spectacularly stupid," West said in an opinion piece, urging Defense Secretary Michael Fallon to step up and explain.
Senior Tories have been attempting a fightback on the issue, with Business Minister Greg Clark telling Sky News "It's been the long-standing policy not to comment on tests of weapons systems and, if that's the approach that you take, I think we have to abide by that approach."
This argument somewhat falls down on the fact that successful tests are regularly reported, including with video of the launches.
Tory head of the Defense Committee Julian Lewis said as much in his intervention early on Monday.
"This sort of event is one that you can't play both ways ... whenever they work, which is 99 percent of the time, films are released of them working," he said.
Lewis said someone should be held to account for the decision.
"I always think with something like this it is better to lay it on the line ... In the end you have always got to assume that something like this will come out," Lewis said.
On Monday afternoon Kate Hudson, head of the Campaign for Nuclear Disarmament (CND) released a statement.
"It has become clear in the last 24 hours that the government acted to cover up a very serious incident involving Britain's nuclear weapons system," Hudson said.
"This calls into question the legitimacy of the Parliamentary vote on Trident replacement that took place in July 2016. MPs should have been told about the safety and reliability issues with the Trident system before they committed to spending £205 billion on a new version of the same technology.
"Both Parliament and public have the right to the truth, which is why we're calling for a Trident Inquiry. Thousands of letters have already been sent to MPs to demand that an inquiry is conducted. It's time for the government act," she urged.
Comment: News report reveals Whitehall covered up 1st-time failure of Trident ballistic missile test near US coast
Update: Defense Secretary Michael Fallon is refusing to answer questions over the June 2016 nuclear missile incident, but it seems US officials have been giving information to the media. CNN is reporting that the incident occurred close to Florida and resulted in the missile, which was not equipped with a nuclear warhead, self-destructing after veering away from Africa and towards the US mainland.