Opening, appropriately enough, the day after the end of this year's Pride, it is hard to think of a more beautifully crafted and engaging plea for love and tolerance than this unexpected gem. The off-Broadway premiere was nominated for the 2010 Drama Desk Award for Best Musical, and it's not hard to see why.

Taking as its starting point the discovery of a journal by a gay American serviceman from the Second World War, what follows is a love story that manages almost entirely to avoid cliché and sentimentality, presenting a credible, and at times downright harrowing, picture of both the horrors of war and the private trauma of men whose personal lives can never be discussed in public.

Despite the seriousness of the subject, sibling writing team Joseph (music) and David (book and lyrics) Zellnik have created a piece that is as light on its feet as it is heavy on the heart. The characters are vividly drawn, the storytelling is swift (sometimes a little too swift) and dynamic, and it is often very funny (I especially loved the trio of male typists who model themselves on the women from Gone With The Wind). There are times when it recalls Kander and Ebb's masterly Kiss Of The Spider Woman (now there's a musical that is crying out for a major revival) in its depiction of men under unimaginable pressure finding solace in unattainable and impossibly glamorous women.

Yank! finds its own voice however, and has moments of brilliance that would not look out of place in Sondheim. For instance, the "Click" tap number in the first half, where hard-bitten photojournalist Artie schools young Stu in how to recognise and hook up with other gay soldiers, is an ingenious piece of musical storytelling, wonderfully choreographed by Chris Cuming.

Despite the attractive swing-heavy score, it is telling that the single most dramatic and enthralling sequence in the entire show – where Stu fights on the frontline but is then incarcerated upon discovery of his journal – is entirely free of music. The showdown between Scott Hunter's anguished Stu and his closeted lesbian military superior (Sarah-Louise Young, absolutely magnificent) is simply riveting drama.

James Baker's superb, uniformly well sung production seldom puts a foot wrong. There isn't a weak link in the cast: Hunter subtly and skilfully delineates the timid, likeable Stu from the modern young man who finds the antiquated journal, while Andy Coxon as the kind, handsome soldier responsible for Stu's sexual awakening (but struggling with his own demons) is heartbreakingly credible. They both have glorious voices. Chris Kiely is equally good as pragmatic photographer Artie, and it was a stroke of genius to cast Young – a cabaret star and stalwart of the Showstoppers improvised musical – as all of the women in the story, from various big-screen sirens, to a supportive mom and a fierce lesbian army officer. Her ability to switch characters and vocal styles at the drop of a hat has seldom been as well showcased as it is here.

It is a shame, given how much great stuff there is in this production, that Victoria Hinton's wooden slatted set looks so cheap. Although material this good and a cast this strong probably don't need much more than a few crates and packing cases (plus Aaron J Dootson's inventive, evocative lighting, and James Cleeve's wonderful band) to make this story work.

Simultaneously gritty and exhilarating, this fine musical engages the heart and the mind in equal measure. Very highly recommended. Take hankies, you'll need them.

Yank! runs at the Charing Cross Theatre until 19 August.