For the past two years, George Shaw has been Associate Artist at the National Gallery. One of the privileges of the post is that he gets the run of the place – of the collection before the public arrives in the morning and the behind-the-scenes activities such as the conservation department. The results of his residency, of his immersion in the works of art history’s greatest names, are now on show in the middle of the gallery in an exhibition entitled “George Shaw: My Back to Nature.”

The exhibition sees Shaw, who was nominated for the Turner Prize in 2011, take a step away from the paintings of the scruffy corners of the Coventry housing estate where he once lived into more bucolic – though deceptively so – realms. The paintings he found himself drawn to in the National are those that shown woods on the outskirts of towns, places where, he says, “something – anything – could happen.” The woods in Giovanni Bellini’s “Death of St Peter Martyr,” Poussin’s “The Triumph of Pan” and Titian’s “Diana and Actaeon” are sites of “transgression and transformation” where Ovidian metamorphoses take place as well as episodes of brutality and debauchery.

So Shaw’s new pictures, which he still paints in the enamel Humbrol paints that as a schoolboy he used for painting model airplanes, are all of trees. Not forest glades though, nothing bosky, sylvan or pretty. His bits of woodland are littered with old beer cans, bits of tarpaulin and the remains of fires. There are no people, though, and the detritus is all that’s left of who knows what scenes that played out in these scrubby edgelands. It was in just such places that he would hang about in as a boy, desultorily killing time with drink and porn mags. In his pictures, lurking behind the banal and quotidian, are some of art’s great themes.

One set of paintings, “Hanging Around,” shows three bare tree trunks and they look at first sight like variations on Constable’s “Study of the Trunk of an Elm Tree” of 1821 – a mix of the botanical and the artistic. Look harder though and the disposition of their branches emerges as three crucifixes and this patch of unremarkable Midlands woodland as Golgotha. Three other large paintings of leaves, tarpaulin and trees send out echoes of the National’s Diana and Actaeon pictures by Titian. Shaw has hidden a clue in the size: his canvases share the same dimensions as Titian’s.

Part of the success of Shaw’s exhibition is this delicacy of allusion. He never points out how he has been influenced by working in the midst of the National’s collection but the sense of it is palpable. Look longer and it becomes clear too that, as in the gallery, there’s every genre on display in his work, from religion to still life, landscape to myth. There’s even a self-portrait, the only one he’s ever painted, which shows him from behind, urinating against a tree. It’s as if he’s the last man standing after the bacchanalia of the night before. His are undemonstrative pictures but potent ones.

“George Shaw: Back to Nature” runs through October 30 2016 at the National Gallery, Trafalgar Square, London.