If an undead creature fails its save against the Necromancy wizard feature Command Undead, and then dies and is raised or rejuvenated (eg a lich coming back from their phylactery) are they still affected by Command Undead?
RAW: likely yes
A dead (neither living nor undead) creature is not a "creature", so it is not a valid target for the ability. However, the ability was used when the creature was a valid target, so this is not a problem. As the ability does not say it ends when the creature dies, it doesn't.
This leaves the question of whether a re-raised creature is the same creature as the previous incarnation. Clearly the body is the same, and at least some of the personality is persistent, as most undead know the languages that they knew in life. Of course, what constitutes a single person is a hotly debated philosophical question, so you should feel free to give your own interpretation, but I think most people will conclude that the re-raised creature is indeed the same creature.
What happens to the CHA saving throw?
When a creature with an INT of at least 12 (such as a lich) is targeted by command undead, it is allowed to repeat the saving throw every hour. I cannot find any rules, SE posts or Sage Advice on objects (as that is what a dead creature is) making saving throws. However, the unconscious condition only specifies automatic failure of DEX and STR saving throws, so I would rule that the corpse makes the saving throw with the CHA score it had while alive (or in this case undead). I think you can also make a reasonable case for a corpse having 0 CHA, giving it a -5 on the saving throw.
At the table: ask your DM
I do believe the above answer to be correct according to RAW, but it would certainly not be unreasonable to rule it otherwise. Therefore, if the DM disagrees, whatever they say goes. Incidentally, this is also RAW. From page five of the PHB:
One player, however, takes on the role of the Dungeon Master (DM), the game’s lead storyteller and referee.