Peter Morgan’s sure-fire hit The Audience imaginatively reconstructs the private weekly encounters between the Queen and her prime ministers over 60 years. The programme usefully reminds us that Queen Elizabeth had already reigned for 14 years by the time David Cameron was born, and has had audiences with a PM (Churchill) who entered parliament in 1900. She may not have any O-levels, as she admits here, but that’s quite some experience of the world. As a constitutional monarch, naturally Her Majesty doesn’t express opinions, argue or try to influence policy: but then where is the drama? Morgan’s solution is to portray the audiences as confessionals, emotional outpourings as much as political meetings. Which may seem implausible, except James Calla­ghan likened his weekly visits to sessions with a psychiatrist.

Morgan eschews chrono­logical order, quite rightly, and looks instead for parallels and echoes. The opening scene begins with a tearful John Major (Paul Ritter) remembering his parents and the Queen tactfully handing him a hankie. He tries to return it afterwards; she politely declines. Not every PM is here — there’s no Heath, no Douglas-Home, no Blair. Edward Fox bravely stepped into the Churchill role 12 days ago, after Robert Hardy cracked some ribs. He is excellent and convincing, even if the only time Churchill was as lean as Fox was after that spell in the Pretoria prison camp.

Helen Mirren is pitch-perfect as the Queen (she has some previous): an utterly believable 26-year-old novice in mourning black; then in an immaculate red suit with diamond brooch and pearl earrings, hands clasped in lap but for the occasional telling gesture; then a stout octogenarian nodding off while listening to Cameron. When younger, she says “Buckingham Pellis” and “unheppy”; “cold” is two syllables, “coeld”. Sometimes she says “nyes”, a hesitant yes that seems to mean: “No, not really, but as con­stitutional monarch, I couldn’t possibly say.” Mirren beautifully captures the humour, the shrewdness, the kindness, as well as the ­melancholy burden and solitude of a life of public duty. She is a wife and mother, but also a ­symbol — “a postage stamp with a pulse”, she observes drily. She emerges with such dignity that the mooted decommissioning of the Royal Yacht Britannia here seems fantastically spiteful and mean-spirited.

Nathaniel Parker doesn’t really look much like Gordon Brown, but I found him riveting: neurotic and tormented, with his Thunderbirds-type lower jaw, compulsive nail-biting and persecution mania. The man who nearly bankrupted Britain is made to appear almost sympathetic. Best of all is Richard McCabe’s charmingly blunt Yorkshireman, Harold Wilson, with whom the Queen really did have quite a rapport, if gossip is to be believed. He jokes that before coming to the palace, he thought a posh building was a public library — just the sort of thing the fake pipe smoker might have said. Watching him wrestling with the sugar tongs is hilarious, and the beginning of his Alzheimer’s is moving. Given that all politicians believe they have the answer to everything, and the right to boss ­others around, it’s impressive how human they appear.

Ah — except Mrs Thatcher. You don’t expect any other ­version of Mrs T in theatreland, it’s true, and you certainly don’t get it. Even the combined talents of Haydn Gwynne, ­Morgan and the director ­Stephen Daldry can summon up only this silly caricature of Thatcher, Thatcher, Milk Snatcher. Gwynne is quoted in the programme as saying: ­“Everything about Margaret Thatcher was antithetical to what I believe in.” Shock, horror — actress professes not to be an ardent Thatcherite. But if Mrs Thatcher was small-minded and unimaginative, with zero interest in the arts, shouldn’t the ­artists at least have more expansive understandings? She is especially wicked here for refusing to impose sanctions on apartheid South Africa. The Queen, meanwhile — heroine of the piece, and as Wilson says, “a good leftie at heart” — cares deeply for the peoples of Africa. To emphasise her saintliness and the wicked racist nastiness of Mrs T, there’s a back-screen video of smiling African children at this point. A definitive “Richard Curtis moment”. You really expect better of Daldry. This weakness apart, it’s a highly enjoyable piece, with beautiful set designs by Bob Crowley. Buckingham Palace is hardly homely, with its louring, monumental grey marble columns; Balmoral is much cosier, thanks to the electric-bar fire purchased from John Lewis in 1965. There are children and animals, too, with Nell Williams wonderful as the 11-year-old princess and a couple of very live corgis, not encouraged to linger on stage too long.

What a great if faintly guilty pleasure this play proves. In times past, the dramatist Peter Morgan would have been locked up in the tower for such impudent lese-majesty, and might have counted himself lucky to have kept his head on his shoulders, but as he showed in his screenplay for The Queen about the crisis that engulfed the Royal Family following the death of Princess Diana, Morgan admires his monarch. And in this marvellous piece, with Helen Mirren once again giving a magnificent performance as the Queen, he penetrates at least some her mystery, with compassion, grace, affection and humour.

When she has fully recovered from her recent bout of gastroenteritis – and one of the many things we learn in The Audience is just how much she hates being ill – the Queen should request a Royal Command Performance at Buckingham Palace. I suspect she would enjoy it enormously. The play is inspired by the Queen’s weekly meetings with her Prime Minsters which have taken place throughout her long reign. No minutes are taken, no officials are present, though anecdotal information has leaked out over the years. Nothing however can be verified. What can be said however is that virtually all the encounters depicted in this play seem persuasive, with the Queen emerging as witty, politically savvy, kind and psychologically astute. James Callaghan, who only has a fleeting appearance here, once said that visiting the queen was like a weekly session with a shrink.

Very wisely Morgan avoids a chronological plod. The play begins with John Major in meltdown about the way his Government is tearing itself apart, then moves back to Churchill then forward to Wilson and so it goes on. What’s remarkable is that Mirren is unobtrusively changed on stage into different costumes and wigs to suggest different periods of her reign and is almost eerily persuasive whatever age she is playing. We also see the Queen conversing with her eleven-year-old self, played by a child actress, making us acutely aware of the fact that from a very early age the young Elizabeth knew what her demanding destiny was to be.

The Audience is often wonderfully funny, with the Queen proving a deft mistress of the verbal barb, but also genuinely moving. She shows real empathy for Gordon Brown when he is clearly battling severe depression and a lovely generosity when Harold Wilson confides that he is suffering from Alzheimer’s. But Mirren also captures the Queen's moments of vulnerability, not least during the break-up of the Prince of Wales’s marriage to Diana, and her sudden anger when she learns of plans to scrap her beloved Royal Yacht, Britannia. Morgan leaves no doubt that what sustains the Queen in her long and sometimes lonely service is her Christian faith and Mirren captures this, too, with stirring simplicity.

Star billing after Mirren must go to Richard McCabe for his wonderful Wilson, initially chippy but quickly smitten by the Queen and deeply touching in his final decline. Paul Ritter is equally affecting as John Major, at one point reduced to tears in Her Majesty’s presence, and there is an icy crackle in the Queen’s dealings with Haydn Gwynne’s steely Mrs Thatcher who enters the royal presence with an extravagant curtsey but then proceeds to queen it over her sovereign. Stephen Daldry directs a pitch-perfect production, with a truly palatial design by Bob Crowley. And yes, there are real Corgis on stage, too.