Temple Church gained worldwide fame when Dan Brown included a major plot point there in his mega-selling novel The Da Vinci Code in 2003, but it has been standing, minding its own business, since the late 12th century. Now it’s home for a short run of Antic Disposition’s Richard III, following a tour of several UK cathedrals – including, controversially, Leicester, where the king's skeleton was reinterred in 2015 after being discovered in a nearby car park.

Every play is a Brexit play. This much we have learnt in the year since the referendum. But in Nancy Meckler’s hands the Globe’s new King Lear becomes the Brexit play – an unpicking of intergenerational responsibility and difference, of philosophies of power and governance, tackling above all that sticky question of what the old really owe the young.

When I say that Matthew Dunster’s Much Ado is revolutionary I’m not talking about the many textual updatings and rewritings, not the lashings of PJ Harvey, nor even the gunfire – weaponised punchlines that cut through the colour and noise of the production.

Can The Tempest open on stage without a tempest – of crashing, shrieking and torment – and thus without what can become five minutes-plus of inaudibility? In Gregory Doran’s 2016 Stratford production for the RSC, revived at the Barbican Theatre, the answer is, as so often, no. Joe Shire, Darren Raymond and Caleb Frederick, playing mariners, have lines to deliver but against giant-wave effects and the supersonic demolition of a ship, they might as well stay mute.

Midway through Hamlet a troupe of actors arrives at Elsinore. Coaching them for his own ends, the prince turns director, delivering an impassioned critique: “O! it offends me to the soul to hear a robustious, periwig-pated fellow tear a passion to tatters…it out-herods Herod: I pray you avoid it.” It’s a philosophy director Robert Icke takes as his own watchword. Out goes declaiming, along with anything demonstrative or self-consciously dramatic, and in its place we get a conversational Hamlet that allows its audience to eavesdrop, forces us to turn voyeur in a contemporary palace of CCTV cameras and hidden microphones.

The effect is brilliantly, dislocatingly fresh. There was a risk that, once freed from the Almeida’s claustrophobic interior into the West End, Icke’s production might lose its uncomfortable intimacy, but instead it merely gains scope in the new contrast between Hildegard Bechtler’s sleek fishbowl of a palace (all sliding doors, hotel lobby-style sofas and chrome accessories) and the wider world that constantly breaks in on television screens that cover the walls. The rot at the heart of the Danish state stinks all the riper for being framed so tastefully – muted visual understatement only broken up by primary coloured flashes of the national flag.

Andrew Scott’s Hamlet who carves such clarity of thought through his soliloquies

But Icke’s updating only starts with design. It’s Shakespeare’s text that gets the biggest makeover in a feat that, whatever your views on the production as a whole, is remarkable for its unerring instincts. This is Shakespeare at length – largely uncut, supplementing the Folio with the Quartos where needed. This isn’t about updating detail or reference (the Emma Rice trick over at the Globe – cut-and-paste contemporary Shakespeare), it’s about finding new inflection in old words, revealing truths that were always there, guiding eye and ear to find new routes through a familiar landscape.

Icke is helped by Andrew Scott’s Hamlet (pictured right) who carves such clarity of thought through his soliloquies, invites us so completely into his play of logic and morality. This Hamlet is no vacillating bore, but a live-wire wit – dry and wryly, self-mockingly funny, puncturing the balloon of his own inflated passions ("Why, what an ass am I?”) long before anyone else can do it. His nervous energy drives the production forwards in erratic bursts of intention, ricocheting off encounters with David Rintoul’s charismatic Ghost/Player King, Peter Wight’s Polonius (a heart-tugging portrait of mental decay) and a tellingly gender-bent Guildenstern (Madeline Appiah).

There are moments of magic: the live-streamed performance of The Mousetrap, video screens projecting the reactions of a court who take their seats in the front row of the theatre itself; the initial encounter with the Ghost – genuinely terrifying; the sexual charge between Angus Wright’s Claudius and Juliet Stevenson’s Gertrude. But there are also some issues.

The dumbshow (scored to a Bob Dylan soundtrack) looks like nothing so much as a Building Society advertisement. Would Claudius really confess his sins to Hamlet (and would he, hearing them, really not shoot him on the spot?); would a modern-day Ophelia really accept so much, so quietly from her lover as Jessica Brown Findlay uncomplainingly does? Both she and Stevenson struggle to find their place in this updating, and Icke’s insertion of a dubious Quarto scene between Gertrude (Juliet Stevenson, pictured below with Wight and Scott) and Horatio suggests a recognition (if not a satisfactory solution) of the problem.

Running at nearly four hours (with an unnecessary second interval breaking the play’s stride just as it should be speeding up), this Hamlet earns every minute of its stage time. Far from a foregone conclusion, the ending reads newly charged as Laertes, softened by Hamlet’s sincerity, has a last-minute change of heart. For one wonderful moment we believe everything could yet be alright, that this will be the bout that ends in a handshake and not a body-count. Those bodies, when they finally come, weigh heavy indeed.

• Hamlet is at the Harold Pinter theatre until 2 September, 2017
• Read more theatre reviews on theartsdesk

OTHER GREAT DANES

Andrius Mamontovas, Globe to Globe. Lithuanian take on the Danish play puts on a frantic disposition

Benedict Cumberbatch, Barbican. Visuals threaten to swamp Shakespeare – and, yes, Sherlock

David Tennant, RSC/BBC. Star looks for life in an infinite space beyond the Tardis

Lars Eidinger, Schaubühne Berlin. Acrobatic Hamlet, outshone by the earth and the rain

Maxine Peake, Royal Exchange, Manchester. An underwhelming production, but Peake is gripping as the young Prince

Michael Sheen, Young Vic. Sheen is riveting as the crazed Danish Prince in Ian Rickson's terrifying psychiatric-hospital staging

Rory Kinnear, National Theatre. Kinnear isn’t a romantic Prince, but an unsettled, battling one in Nicholas Hytner's staging which is modern, militaristic and unfussy

Overleaf: Robert Icke's dazzling career so far

Nature’s germens tumble all together rather readily in more recent operatic Shakespeare. Following the overblown storm before the storm of Reimann’s Lear and the premature angst of Ryan Wigglesworth’s The Winter’s Tale, what's rotten in the state of Denmark rushes to the surface a little too quickly in Brett Dean's bold new take on the most challenging of all the tragedies.

Tomorrow Britten’s opera A Midsummer Night’s Dream will begin a short run at the Snape Maltings, Suffolk in a new production directed by Netia Jones and conducted by Ryan Wigglesworth. It will mark the high point of the Aldeburgh Festival’s summer celebrations half a century on from the opening of the Snape Maltings concert hall. It is therefore more than a happy coincidence that back in 1967 the "Dream" was aired as part of the hall’s maiden season.

The Globe’s artistic director Emma Rice has made no secret of her desire to go out with a bang, in this, the final season of her brutally truncated tenure at the company. With this Twelfth Night she stages a departure with bells (and whistles, and disco-balls, electric guitars, congas, Sister Sledge, and yes, a whole rig of lighting) on – a neon-bright, two-fingered salute to the board that forced her out.

There may never have been a time when Shakespeare’s Richard III did not have contemporary relevance, but surely never more than it does right now. And it’s to the credit of director Mehmet Ergen that this production doesn’t go to town on it, but instead leaves the audience to make its own connections.

“Everything in extremity”. That announcement that the Capulet party is about to begin could just as well serve to describe Daniel Kramer’s Romeo and Juliet as a whole. Opening the Globe's new season, it will provoke reactions as conflicting as the play’s warring families.