I know some people think it's silly to be hung up on continuity. Regardless, I'm one of those who is hung up!
From what I can tell, the Superman Reborn arc was a mini Crisis and brought back most of pre-Flashpoint and erased most of New 52. Bart was lost somewhere and found his way back home.
From there? My guess is he searched out for one of his best friends, Tim. Bart touched Tim and caused the same thing to happen to Tim that happened with those Wally touched. Tim began getting memories back of Connor & co. That's why Tim said "not without Connor", because Bart caused him to remember everything.