Speculation has centered on the rumor that Billie had dental work done, to which she had not fully adjusted.
But even if that was so, it doesn't splain why Rose would be having a problem as well. And as busy as she was (as a member of Torchwood in another dimension) in trying to save all of the parallel dimensions from the oncoming threat of Davros, I don't think she had time to pop off down to the dentist to have work done on her teeth.
However! I think I do have a good, shimple shplainin for why she shounded sho shtrange.
(Shorry about that, Chief!)
The Doctor had warned Rose that the walls between her dimension and that of her original home had to remain sealed forever. Otherwise they would shatter and destroy both worlds. But I think Rose and the scientists of Torchwood in that other world found a way so that Rose could teleport back and forth through the dimensional vortex without shattering the barrier between both worlds. Being a parallel world, it was probably along the same principles discovered by Quinn Mallory on 'Sliders'. (For alls I know, the "Q-Ball" of that world developed it as well!)
So Rose is able to make the jump, to slide back and forth from her Toobworld to the main one. (I still think I can salvage 'Doctor Who', even with all of its discrepancies, so that it can remain a part of Earth Prime-Time.)
However, as she is doing this without the protection of a vehicle or even padded gear, I think she's getting buffeted so badly by the transition that it's knocking her for a loop. Her fillings are literally being shaken loose by the ordeal.
And that's why she's got that lips problem.
I already know we won't see this, and since this is all theory, it's not a spoiler, but if you just want to make sure, look away now.......
I think that outside the view of the home audience, the Doctor used his sonic screwdriver to eventually fix her teeth.
The Doctor as an orthodontist. Who knew?