I just had to check, but these days it's "Live from the BBC..."
That said, last week's opening montage began with a drone shot of big hanger studio at Elstree that was clearly not Television Centre ("TVC"). I think viewers who are even vaguely interested in where the show is shot know it's not in the old building any longer.
As you say, TVC is now a studio for hire, and hosts many of ITV's live daytime shows in one of its stages. ITV is moving into a nearby building that was once owned by the BBC in a nearby complex. ITV sold it's own facilities, The London Studios, on the Southbank of the Thames, also for housing development. But BBC shows are also still made at TVC too - Graham Norton for one. Ironically, his show was formerly shot at ITV's The London Studios... All the old TVC offices are now apartments and there's a branch of Soho House in there too. But BBC Studios - the BBC's commercial arm - still has offices there.
Adam