Every week of her 64-year reign, Queen Elizabeth II has met with the elected Prime Minister of the day to discuss the routine business of politics in Parliament. To date, she has met weekly with 13 Prime Ministers, from Winston Churchill to our current, unelected PM, Teresa May. The meetings are underpinned by a hard-fought point of principle: the UK monarch is the entirely neutral Head of State, entirely denuded of real political power and only fulfilling ceremonial and official duties.

The relationship between monarch and Prime Minister has been a source of intense gossip, rumour and imaginative engagement. We know, for example, that Queen Victoria loathed Prime Minister William Gladstone, grumbling that he addressed her in hectoring tones, as if she were 'a public meeting'. She much preferred his rival, the charismatic, obsequious Benjamin Disraeli, who secured her affections by making her Empress of India in 1876. Queen Elizabeth II, like her predecessors, has maintained a public aura of studied, dignified neutrality; she never gives interviews and her political opinions and her relationships with various PMs, although the source of fervid speculation, are officially unknown - albeit strongly apparent from some of her actions and Christmas speeches.

The relationship between Elizabeth II and the UK's first female Prime Minister, Margaret Thatcher has been a source of particular fascination, especially as rumours abounded for years that the two women loathed one another. In 1986, an article in The Sunday Times reported off-the-record remarks that the Queen was deeply unhappy with Thatcher's refusal to support sanctions against South Africa, intended to bring an end to the racist brutality of Apartheid. This moment signalled the adoption of a far less deferential attitude to the monarchy by the press, which culminated in Elizabeth II's annus horribilis speech in 1992, when the marriages of three of her children collapsed - after fevered press speculation and gloves-off paparazzi harassment - and part of Windsor Castle burnt down.

Moira Buffini's 2013 play imagines the younger and older Queen - identified only as Liz and Q - meeting with the younger and older Margaret Thatcher - Mags and T. Both the Queen and Thatcher were notoriously indifferent to the theatre in real life, possibly because public life involves so much performance, but Buffini has the women meeting on an unidentified stage in an apparently empty theatre. This hints at the loneliness of power, particularly as the two women recount moments of intense stress when they kept their heads as all around them lost theirs. Two jobbing actors, who play all of the other roles from Neil Kinnock to Nancy Reagan, attend them. The play revels in its meta-theatricality : the characters are aware they are on stage, there are many self-conscious asides to the audience, and the male actors bicker over who will perform the best parts, while being unceremoniously ordered about and humiliated by Mrs Thatcher when they try to draw attention to the suffering inflicted by her socio-economics policies.

The play is roughly chronological, running from Mrs Thatcher's 1979 election win through to her unceremonious fall from power in 1990. Major events covered include the 1979 Commonwealth Heads of Government meeting, the 1982 Falklands War, the 1984-1985 Miners' Strike, the 1984 Brighton Hotel bombing, the 1985 Bridgewater Farm Riots in Tottenham, and the fall of the Berlin Wall in 1989. It's delightful to see the Queen and Thatcher sparing with the utmost civility and graceful, even-tempered ferocity, with the Queen demurring at Thatcher's callous treatment of the miners, lending considerable moral weight to the fight against Apartheid, and triumphing over a diplomatically inexperienced Thatcher at the Commonwealth meeting in Lusaka, Zambia. The deferential Thatcher, who idolised the monarchy, is baffled by the Queen's attitudes yet determined to impose her will.

Buffini's great skill lies in her refusal to present either woman in limited, one-dimensional terms. Mrs Thatcher is terrifying in her determination, focus and ideological fervour. This is a woman unshaken in her absolute self-belief, suppressing her emotions with steely self-denial, espousing and imposing her neoliberal dogma with swivel-eyed authoritativeness. Anybody who attempts to question or inhibit her will is simply branded a traitor or enemy within and summarily dismissed. But we witness her horror in the immediate aftermath of the Brighton bombing - although she rebuffs the Queen's genuine concern for her safety - and her grief when her mentor Airey Neave was killed in an IRA bomb attack outside Parliament in 1979. The Queen is similarly self-controlled and contained, disguising her real feelings in order to embody the studied neutrality her constitutional role demands. Although the Queen often comes across as warmer, more affable and emotionally attuned, Buffini sharply reminds us of her immense wealth and her life of unearned privilege and comfort.

The four leads are uniformly excellent, with strong support from Ian Recordon and Jonathan Wober as the much-put-upon actors. As the younger Queen, Helen McCormack is cordial, gleeful and enthusiastically frank, presenting the Queen as curious, ebullient and more than a match for the Iron Lady. Alison Liney presents the older Queen as more wounded and guarded, but still with a firm sense of her duty and her moral revulsion at Thatcher's ideological emphasis on the individual at the expense of society. As the younger Thatcher, Julie Arrowsmith delivers a pitch-perfect impression of that chilling, stentorian voice, giving an impressive performance of barely restrained fervour as the emotionally repressed, dogmatically driven and supremely self-assured Prime Minister. As the older Thatcher, Anne Connell is still driven, fixated and ruthlessly strong-minded, but tempered by the hurtful experience of violence, betrayal and her ruthless ousting as Prime Minister by her own party. Her opening speech is delivered in such a perfect rendition of Thatcher's voice that I initially thought she was lip-syncing.

Jude Chalk's impressive yet simple set design - a near-empty space, with four stylish chairs, a tea trolley, a giant Union Jack on the floor and a crown floating above all - perfectly suits the play's self-aware theatricality and ensures the focus is deservedly on the wonderful performances. The costume design, by David Taylor and Kathleen Morrison, is period perfect - Ian Recordon looks gorgeous in full regalia as Nancy Reagan. This is, in many respects, a deceptively simple play - a small company on a simple stage - but it is richly complex in emotional and performance terms and Martin Mulgrew directs with assurance and has elicited striking performances. There is a great sense throughout that the actors are thoroughly enjoying themselves. He must be particularly commended for having secured such uncanny impersonations of Thatcher and the Queen from his four lead actors, but he has allowed the actors to move beyond imitation to imbue their performances with an impressive breadth of sympathy and liveliness. There was also a fantastic 80s soundtrack. On a couple of small occasions, the movement around the stage was a little stilted or lacked purpose, and some of the transitions between scenes lacked fluidity, but overall, this is a very strong, enjoyable production of a thought-provoking and very amusing play, delivered by an impressive cast.