There was actually a line cut from The Christmas Invasion that explained the estuary accent, where it’s shown that part of Rose’s personality has infected the new Doctor’s character.

‘It’s a shame they cut that!’ he exclaims, sounding genuinely peeved. 'That was the other reasoning behind it as well, this idea that Rose had imprinted on him and I think it made a nice sense, and it feeds into that whole close relationship that they’ve got and how that was unfolding.’

—  David Tennant on not using his own Scottish accent in DW (2006)