[JAS84]

There's nothing more to tell. They just used the news ident for the rest of the night because David Tennant was no longer The Doctor. The next morning, of course, they reverted to normal idents (the Christmas special that year was a two parter which aired on Christmas Day and New Year's Day).

There'd have been no issue with that had it happened after they stopped airing Christmas idents on New Year's Day. This year's Who Christmas Special will be a debut episode for Ncuti Gatwa's new Doctor, after next month's three specials with Tennant, so if the ident did feature the Doctor, it wouldn't be rendered out of date after it airs like the 2009 idents were.