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"I am speaking to you from the Cabinet Room at 10 Downing Street…" With those famous words, Prime Minister Neville Chamberlain announced the beginning of the Second World War. This simple, effective two-hander takes us inside that Cabinet Room, witnessing the nervous moments leading up to Chamberlain's historic radio broadcast. He's lonely, but he's not alone, and there are a couple of unexpected twists to this solidly informative historical drama.

The biggest surprise is that there's a live singer on stage – performing crowd-pleasing classics like The White Cliffs Of Dover, and a smattering of less familiar tunes. The pretext is that the BBC are playing light entertainment while they wait to cross to Downing Street, and there's a satisfying irony in this "light and positive" backdrop to the preparations for war. But this is really just an excuse for actor Colin Alexander to show off his singing skills, and worthy of showing-off they certainly are: instantly nostalgic, richly resonant and musically strong.

The songs are always welcome, but the heart of this production lies in the spoken dialogue. Chamberlain is joined in the Cabinet Room by an assistant and confidant, also played by Alexander; as the two men wait nervously for the cue to go on air, they discuss the sequence of events that's brought them to the eve of war. The level of detail is pitched just right – enough that you'll learn something new, not so much that it feels like a lecture – and the factual background is neatly woven into Chamberlain's human story. Playwright David Robinson's script is historical drama as it ought to be done.

Playing Chamberlain, David Leeson presents a morose, defeated, even sulky figure, crossing the stage with heavy footsteps and head always slightly bowed. The play advances and supports the theory that Chamberlain was "the right man at the wrong time", and his anguish at the failure of his hopes for peace is almost tangible. But the script also deftly touches on the political machinations surrounding the declaration of war – a topic we rarely hear about, and which this production wisely allows to speak for itself, without reaching for tempting but clanging modern parallels.

There's one lapse into a textbook recital of dates and places, and the device of having characters explain their inner thoughts directly to the audience isn't as subtle or natural as I'd have liked it to be. But there's a genuine tension in the unremitting countdown to 11 o'clock, the precise hour when no undertaking is received and consequently, the country is at war with Germany. At one point, I realised that my pulse was racing – a genuine achievement for a quiet play involving two men in tailcoats, when everyone already knows how it ends.

Chamberlain is an understated and effective show which offers a thoughtful portrait of a man who's been harshly judged by history. And with bellicose rhetoric once again ringing out from across the Atlantic, there's an added resonance to its quiet analysis of the forces that drive us to war. It's nostalgic, yes; old-fashioned, perhaps. But I believe it's a piece for our time.