The Milan show and upcoming ones in the contemporary space - “The Art of Bollywood” comes in the summer, while autumn is given over to classy advertising since the 1960s - demonstrates a sure populist touch. This from a man you might expect to be somewhat stand-offish, not to say elitist. His background is impeccably officer-class - Eton and Oxford, followed by a lengthy stint at the Courtauld Institute, a short spell at the British Museum as a coins and medals specialist, and then his key role prior to the V&A as director of the National Museums of Scotland from 1992-2001. During which time he oversaw the construction of a controversial new extension, the exemplary modernist monument of the Museum of Scotland in Edinburgh by Benson and Forsyth. Changing the image of such a previously fusty institution turned out to be a dry run for the V&A. After that, Libeskind’s Spiral holds no terrors for him.

Jones himself is quite clearly inclined towards the contemporary, for all that the first big move in his ten-year reshuffle will be the creation of a huge new suite of medieval and Renaissance galleries in the south-east corner of the complex, balancing the new British Galleries in the south-west. The day I meet him, he has finally pushed the antique Alan Borg chairs to one side in his office, and taken delivery of £600 worth of state-of-the-art American “Aeron” office chair, as seen on the BBC’s “Question Time”. Not that I see him sitting in it. He is up and out of the room, jangling his bunch of museum keys like a genial gaoler. We are going to walk the walk through the building he has come to regard as a mini-city.

The tour is done at breakneck speed. I thought I knew the old place pretty well, but Jones soon has me somewhat disorientated. The V&A was built piecemeal from 1860 to 1909, and has suffered numerous equally piecemeal internal alterations since. Previous masterplans - most notably by architect Sir Michael Hopkins in 1985 - got nowhere. No wonder the architectural historian Sir Nikolaus Pevsner described the place as having “an almost naughty looseness”. We take short cuts through locked doors and along private corridors. We walk on high mezzanines closed to the public. We look through hatches into forgotten roof vaults. There is a reason for this magical mystery tour. Jones’ new masterplan for the V&A, drawn up with architects Metaphor, involves opening it up, reinstating long-lost views from one gallery to another and, where possible, from floors to original roofs.