"What country, friends, is this?" Viola's first words have rarely felt so resonant and, in Emma Rice's joyous, giddy and yes, not-for-the-purists-blah-blah Twelfth Night, they open the show. Not "If music be the food of love…" but "What country, friends, is this?" Shouted into the summer's night on the South Bank, almost a year on from a referendum that split the nation, it's a question that concerns us all. In no time at all, ours has changed almost beyond recognition. "What country, friends, is this?"

Admittedly, watching Rice's Twelfth Night – the last show she'll direct on the Globe's main stage – the politics aren't immediately apparent. It is, first and foremost, an absolute hoot; arguably less Twelfth Night itself than a Twelfth Night-themed revue. The play is perfectly present and correct, but it is freely embellished and fulsomely overdone. There are pop songs and dance routines, sometimes even swerves into full-blown musical theatre as composer Ian Ross sets Shakespearean verse to song. We get prat falls and variety turns, big wigs and drag queens, and, as in Kneehigh's work, clownish caricatures instead of credible characters. Sniff if you will. It's a complete blast.

We start on the S.S. Unity – a camp-as cruise ship helmed by the glorious Le Gateau Chocolat, a big, bald, bearded drag queen with eyelashes like diving boards, surrounded by naval officers in spic 'Top Gun' whites. That party boat comes a cropper with a sudden lurch, not only spilling sailors this way and that, but splitting Anita-Joy Uwajeh's spirited Viola from John Pfumojena's Sebastian.

Ilyria, here, is the heart of the Highlands. Joshua Lacey's preening Orsino is a mulleted Monarch of the Glen, jigging like a salmon swimming upstream. He suffocates Annette McLaughlin's starchy Olivia with the best Athena's shelves have to offer: red heart balloons, cheap roses and mixtapes of his own making. At her place, Katy Owen's tweedy little moustachioed Malvolio, a finicky Welsh domestic sergeant, perfectly sexless, toots his whistle and scolds his staff, keeping up standards that Tony Jayawardena's Toby Belch and Marc Antolin's Andrew Aguecheek are intent on letting slip.

Clothes maketh the show – as they should in Twelfth Night. With its mourning dress and cross garters, it's Shakespeare's fashionista moment. Lez Brotherston's costumes extend that thread. We get locals in woolly jumpers and trad tartans, and sailors in uniform. In a household of austere dressers, the dandies run riot: Belch in his baby blue tux, Aguecheek in a pink Pringle jumper, Carly Bawden's Maria forever flashing the pants beneath her maid's dress. Le Gateau Chocolat's Feste, of course, dazzles in a gold-glitter gown.

They are, of course, creatures of the night. "To go to bed after midnight is to go to bed betimes," as Aguecheek says, and with every tequila shot they slam, the night lights up: the pale moon turns green, the night-blue sky pings pink. This is, at times, a celebration of all that nightlife offers – sauciness, silliness and self-expression – at a moment when London's seems to be seizing up. These free-spirited fops are all about fun, and their prank tips too far into cruelty.

When Owen's Malvolio emerges, not in beige shirt and brown trousers, but canary plus fours and a striped sleeveless vest, her quivering smile betrays initial discomfort. Soon, he lets loose, boogying by himself: "Let me enjoy my private." That they don't, that the trio ridicule his new off-beat look, tilts their snobbery into spitefulness, even fashion into fascism. I've never known a Twelfth Night slide our sympathies from one party to another so delicately. The play's subtitle looms large: Or What You Will.

Anyone still out to spot internal Globe politics will find plenty. Here are conservatives and bohemians at odds with each other – both parties capable of making the other feel small. That's Brexit too, of course: hipster elites and country squares. And Rice offers an antidote: the S.S. Unity is all for one, its lifebelts spelling out 'In Love We Trust,' while Sister Sledge's "We Are Family" shimmies throughout. Come the end, out come the white flags: a semaphore dance routine that signals a surrender.

Audiences should take note: drop your guard, forget 'proper Shakespeare blah blah' and just enjoy. This Twelfth Night's a delight: a cheek-ache from start to finish. Great art? Probably not. Great fun? You bet.

Twelfth Night runs at Shakespeare's Globe until 5 August