The first situation's not a problem; the rest of the party can just attack the skeletons that the Cleric failed to turn, since there are some. It'd be a problem if the Cleric turned all the skeletons, and so then the party just sorta stood around doing nothing for 6ish seconds before the skeletons fled. The second situation is more of a legitimate case of this problem.
The solution here, in my experience, is for you and your players to learn the metaphysics of the D&D world so that you understand beforehand what sorts of actions make sense and which don't. Combat is not simultaneous and turns are weird, whatever the developers might say about them. They work and are useful for creating a fun combat system that's simple to understand and easy to play. They do not and cannot model reality in a manner consistent with expectations of simultaneity. If you want to pretend that combat turns are actually all happening at the same time somehow, you are going to run into a lot of problems like this all the time. The solution is to give up, accept that turns aren't simultaneous, and learn the sorts of tactics that work in the non-simultaneous world or, if you'd prefer, to use a different game system that can handle those expectations.
What you can't do is make some minor changes-- a couple house-rules and finer grained initiative segments, maybe some homebrewed durations that actions take-- and end up with something that works well. People have tried that lots of times, and by the time the system actually does work pretty well, both at being turn-based and at being simultaneous, you have something that resembles Phoenix Command a lot more than it resembles any edition of D&D.
If you are looking for a different system to use that can do this, look for games that have a much more zoomed-out view of combat, so that parties outline a general course, then the result is figured out, then what happens is narrated, not for individual actions but for much longer segments of a fight or even the whole scene. Taking the emphasis off of the details of combat makes it easier for designers to avoid invoking a turn system.
If you are looking to adapt to the nature of the turn system, consider preferentially targeting enemies whose initiatives are between yours and your next party members, and avoiding spending resources against enemies whose initiative is just before yours. If you neutralize the enemies between you and the next party member for the round, then they can focus on neutralizing the next ones. This isn't always the best idea, because sometimes one enemy is much more dangerous than its companions, and ignoring those companions would be a better choice. You should be aware, though, that if you target an enemy whose initiative is later than the rest of your party's, any effect that doesn't actually kill or otherwise completely neutralize the target semi-permanently may well be wasted.
If you must try to make a house rule to handle this, consider changing the effects in question, rather than mucking with the initiative system. Make things that would happen at the start of an opponent's next turn instead happen now, and have them consume the target's next turn (but not change their initiative). This makes the abilities more powerful, and makes their use against enemies further away in the initiative order preferential (since they will be out of the game for longer, like normal, but also you don't have to wait for the stuff to trigger anymore), but will help a bit with this particular problem.
