\Why no, Lord Uhtred does not take possession of Bebbanburg. There was no plot reason that Uhtred didn't take Bebbanburg, despite author's handwavium trying to explain it. The obvious reason is that author can't let go of this device to keep Uhtred on the playing field, under the dominion of (some) hated others, rather than becoming the power in his own right he couldn't avoid being, if he sat as lord of his own domain fortress.
That latter development would make for a lot more interesting and new stories than the repetition of what we've already had. Since in terms of the series, we're just at the turn from 10th to 1Ith century perhaps having the fictional Uhtred taking possession of the real Bebbanburg would screw the history too much; the Vikings didn't destroy the Saxon fortification until 993.
The time has arrived to retire Uhtred, who has already survived significantly beyond the average span of a warrior who maintains his leadership as part of the shield wall.