This wouldn't work, due to the wording of the spell. The revivify might work, but at the start of the lich's turn he would still become a zombie.
If we're using the rule as written, which I tend to do, I would say this would not work.
A humanoid killed by this spell rises at the start of your next turn
as a zombie that is permanently under your command, following your
verbal orders to the best of its ability.
I emphasized killed because this is the most important part of this question. Your assumption is that the creature must be dead to become a zombie at the start of the lich's turn. The spell only specifies that they must have been killed, not still be dead.
Revivify could resurrect that person, but they would be turned into a zombie at the start of the lich's turn anyways, because they were killed by the spell, which is the only trigger that the spell specifies other than "at the start of your next turn".
Also, there seems to be confusion on the wording "rises"
The wording of the spell also states that the humanoid will "rise" as the start of the caster's next turn. I don't believe this means they need to be dead to rise. For example, you would assume a creature with no legs would still be affected by "Irresistible Dance", even though the spell states "shuffling, tapping its feet, and capering".
In summary: Revivify doesn't counter the fact that they were killed, and that is the trigger for the spell (not being dead)
Below is only flavor, because as a player I enjoy hearing the in game reasons, or flavor, behind every action.
This is not information to support my answer, this is only to give you an idea of how you might explain this in game to the players in character.
As a DM, explaining this in game might be tough, but if the cleric were to try to revivfy them, I would flavor it as "The creature's soul is already gone, and under the lich's control. You can try, but it would be in vain as the lich has control of them." I wouldn't want to penalize the character for an action or a spell slot just for trying. I might even award inspiration, but still advise against using the spell slot. I would also flavor this as the cleric knowing the fate of the peasant as he kneels to attempt, having probably revived someone successfully before. This time would be different, for them.
In game, if they went through with the resurrection after my warnings, I would have the humanoid come back to life, thankful but then grotesquely shift into a zombie anyways. It would look painful and terrible.