Gorton and Denton by-election was run on a very compressed timetable, and the basic timings were:
22 Jan - Andrew Gwynne announces his resignation
28 Jan - Manchester City Council calls the by-election
26 Feb - Polling day
If there's a vacancy in Bootle tomorrow, Burnham still isn't getting back to Parliament until the second half of June at the earliest. September might be more likely.
And he would still have to get his new constituency to elect him. Not guaranteed.
But let's say he gets back in late June or early July. Then you have to have the whole rigmarole of a leadership contest, unless everyone else agrees to let him have it uncontested. The NEC could compress the timetable so the contest runs through the summer and the new leader is announced at conference in September.
It's just going to keep this zombie premiership going for months, unless either Starmer gets sick of it and resigns of his own accord (not likely), or the PLP discovers how to stage a coup (for context, they got 80% of MPs to vote no confidence in Corbyn and he just refused to go).
This is how he staggers on, partly because Labour has this antiquated rulebook in place of the 1922 Committee, but mostly because all the other contenders are just as useless.