Miss Atomic Bomb, at the St James Theatre, is set in 1952 when nuclear bombs were being tested near Las Vegas, making the area a tourist attraction.
Mark Shenton, in The Stage, said it is "a bomb in almost every way", awarding it just one star.
Daily Mail reviewer Patrick Marmion said it "never really detonates".
'Likeable but lame'
He awarded the show two stars, saying there are "too many story strands", but notes that the cast "toil and labour and travail".
Former Doctor Who star Tate, playing fashion designer Myrna, "makes the best of toe-curling gags" but her comedy skills are "underused".
Shenton said "the satire is lame, the songs desperately derivative", adding: "Though it has been in development for some five years, what has fetched up at the St James still feels like a very early draft."
Despite its original story and score, "its chaotic and formless comic structure lacks purpose, polish and point", and Tate's "impossible accent that veers from Southern drawl to Australia", he wrote.
The Guardian's critic Lyn Gardner described Miss Atomic Bomb as "mildly likable but lame".
Also giving it two stars, she wrote: "There's a kernel of something potentially intelligent and interesting here, particularly on the themes of personal and political betrayal.
"But the show goes for a 1950s screwball comedy style and misses it not just by a mile but by an entire exclusion zone."
Tate stars alongside Simon Lipkin, with whom she appeared in Stephen Sondheim's Assassins at the Menier Chocolate Factory.
He plays hotel manager Lou, who is looking to cash in on the tourist boom as the city waits to see who will be crowned the beauty queen in the Miss Atomic Bomb pageant, based on contests that took part at the time.
Dean John Wilson, set to star in Disney's Aladdin in the West End later this year, stars as soldier Joey, alongside Florence Andrews, who plays Myrna's friend, farm girl Candy.
Holly Williams, in the Independent, said the musical has an "audaciously daft plot" with a "cartoonish cast".
She described it as a "slick show, with plenty of pizzazz and some fun performances", but concluded it was "thin fare".
Michael Coveney, writing for What's on Stage, said the musical is "curiously flat, unfunny and flair-free", also giving it two stars.
"The songs have no zing or provenance and sound as though they've been made on a computer, or compiled on a 'how to write a song' study course; there's not a single moment of lift-off, melodic surprise or harmonic twist from start to finish," he writes. "Aha - maybe that's a result of the radiation emanating from the Nevada desert."
But Dominic Cavendish writes in the Telegraph that, although "bonkers", the musical is "enjoyably macabre", giving a three-star review.
"Those with sophisticated tastes should probably run for the hills and not look back," he said. "But even though it's overlong, I have to confess to experiencing a steady rumble of grudging admiration that often erupted into outright laughter."
And Benjamin McDonald, writing in the Gay Times, gave it four stars, saying it "provides a much needed explosion of originality in London's theatre district".
He praised Tate, saying she "proves she has the strong vocal ability to match her impeccable comic timing" but says Andrews steals the spotlight with her "magnetic charm".
The musical, co-directed by Bill Deamer and Adam Long, is at the St James Theatre until 9 April.