What happens when a play is accidentally awarded a touchdown?

It depends on exactly what is overturned, but in various leagues and conferences replay typically has a specified set of things that can be overturned.

In the case of a fumble, the general rule is that if there is "clear recovery", including "recovery" by the out of bounds line, then the ball is given to the team that would have recovered it; but if there is not clear recovery, i.e. the ball was blown dead and it is unclear to the replay official who would have recovered it, the call is not usually overturned.

For example, in the 2019 NCAA Football coaches manual, they mention this for fumble recovery:

d. Loose ball ruled dead (Rule 4-1-2-b-2), or live ball ruled dead in possession of a ball carrier, when the clear recovery of a loose ball occurs in the immediate continuing football action.

1. If the ball is ruled dead and the replay official does not have indisputable video evidence as to which team recovers, the dead-ball ruling stands.

In this case, from other comments on Twitter it looks like an LSU player actually recovered the ball prior to it going out of bounds, so LSU would have been given the ball at the point where they picked it up and not been permitted to advance it; however, given they would have picked it up off of the goal line, it would have been ruled a touchdown anyway.