No.
You make as many saving throws as you have effects.
PHB p.279
Saving Throws
- End of Turn: At the end of your turn, you make a saving throw against each effect on you that a save can end.
- Choose Order: Whenever you make a saving throw, you choose which effect to roll against first, which effect to roll against second, and so on.
Sleep causes two status effects. That means two saving throws. Furthermore, there's no order in which the saving throws must be successful. If an Unconscious, Slow creature succeeds only on its saving throw against Slow, it's still Unconscious. Its next turn, it need only make one saving throw against the Unconscious effect to begin the following turn with full movement.
It doesn't hurt roleplaying to do it this way either. For example...
- (Player's turn, Sleep attack roll hits) "You see the monster's legs become wobbly, as though on the verge of passing out."
- (Monster's 1st turn, fails Slow saving throw). "The monster takes a few steps towards you, but then its eyes roll back as its legs fold under it. It's snoring before it even hits the ground."
Possibility 1:
3. (End of M's 2nd turn, fails Unconscious saving throw, saves against Slow) "The monster begins to twitch rapidly, as though running in a dream."
4. (End of M's 3rd turn, saves against Unconscious) "The monster's eyes pop open. It's fully alert, and it looks pissed."
OR possibility 2:
3. (End of M's 2nd turn, saves Unconsicous, fails Slow) "The monster slowly opens its eyes. It's awake, but still moving lethargically."
4. (End of M's 3rd turn, saves Slow). "The monster's finally shakes off the last remnants of sleep."
OR possibility 3:
3. (End of M's 2nd turn, saves Unconscious, saves Slow) "The monster's snores end abruptly. Its eyes jerk open, and it looks unwaveringly at you."
OR possibility 4:
3. (End of M's 2nd turn, fails both). "The monster continues to take each breath slowly and deeply."