Technically, if the last 20 years have changed there's no reason any of the same people should be on the Enterprise.
And in any mirror universe there's no reason the exact same people should have even been born.
But the timeline only diverged 22 years earlier, so that argument would only apply to people under 21 or so (allowing 9 months for gestation). More so for younger people, since the timelines would still be pretty close to start with and diverge gradually.
The main thing that bugged me about "Yesterday's Enterprise" was that the Galaxy
class was in use at all. After all, that class of ship was meant to be a peaceful deep-space research vessel, a university village in space loaded with civilian scentists and crewmembers' families (because the ship was meant for really long-term exploration missions of up to 15 years away from home, and few would be willing to leave their families for so long -- although later TNG producers completely ignored this intention). The saucer section was designed to house the civilians and separate to keep them safe when the drive section went into battle. And the Klingon war in the alternate timeline started over a decade before the first Galaxy
ships were launched, enough time that there should've been some major redesigns to gear the ship more for combat.
Of course, they didn't have the budget to build a whole new ship miniature or new sets for the episode. But maybe they could've used just the battle section of the 6-foot miniature and shot the bridge scenes in the battle bridge set -- with maybe the sickbay or crew quarters set redressed as a wardroom to take the place of Ten Forward for Guinan's scenes.