Pretty much as Cows Getting Bigger calls it. BA bashing is probably quite an easy sport and if the clip has been edited that raises alarm bells. Clearly the YouTube poster is not impartial.
They should have vacated when instructed, but the ATCO should not have vacillated. The part about taking a phone call is probably a badly worded response when put on the spot, but didn't help.
They should have taken a leaf out of the blue-now-purple brigade and declared a "small technical problem, we just need a moment to investigate". Used to great effect when lined up on the active without being fully ready and with aircraft now barrelling down the ILS, having declared ready for departure.