I think these predictions are mostly on the "Donna is Romana and Wilf is secretly a Time Lord" side of the likelihood scale, but it's fun to speculate none-the-less.

On my second watch through I caught Nardole saying somethings which suggest he's sticking around because he has a crush on the Doctor, which I really hope isn't the case, because, well... If his purpose turns out to be guarding the vault while Bill and the Doctor go traveling, then I'll be more than happy as well.