The GM decides when they make a new save
The game has several definitions as to when a creature makes a new save to end an effect:
Effects that call for a saving throw at the end of your turn will explicitly say so, such as Glitterdust, Mental Block or Coward's Lament.
Those that are at the start of your turn will also say so, such as Greater Command, Web Cloud or Death Knell Aura
Or will simply say exactly when those creatures get to make a save, such as Entangle saying they must save at the end of the caster's turn.
Other spells, like Hold Person, have a specific mechanic in place (for Hold Person, a full-round action) so the creature can make this save to end the effect.
For Turn Undead, the effect says that the creature gains a new save each round:
Intelligent undead receive a new saving throw each round to end the effect.
The Combat Round
If we look up the definition of a Round in the game system, we have the following:
Combat is measured in rounds. During an individual round, all creatures have a chance to take a turn to act, in order of initiative. A round represents 6 seconds in the game world.
So if we rule that it's at the start of a creature's turn, then it may be anywhere between 0 and 5,99~ seconds, depending on their initiative order.
If we rule it's at the end of a creature's turn, then it could be a little closer to 6 seconds, but still not exactly 6 seconds, regardless of their initiative order.
If we rule that it's at the end of the round, after 6 seconds have past, then the ability's text should say "after each round".
Confusion has a very specific wording regarding when the effect happens, but also makes a difference between rounds and turns:
at the start of each subject’s turn each round to see what it does in that round.
And again, we have to remember that turns are simultaneous, creatures aren't exactly waiting for each other to do something in combat. With that in mind, it's safe to assume that if this effect was at the beginning of a creature's turn, then it would be at the start of those 6 seconds, and that wouldnt be a round by the definition of the game term.
For effects that call for a "full round", we have a very specific rule about this on the combat chapter, which could be applied to Turn Undead:
When the rules refer to a “full round”, they usually mean a span of time from a particular initiative count in one round to the same initiative count in the next round.
Effects that last a certain number of rounds end just before the same initiative count that they began on.
So, if a GM decides to allow a new save at the end of the character that caused the effect, they will have some backup on the rules.
James Jacobs, Paizo's Creative Director, when asked about this, said that he would rule it to make a save at the start of the undead's turn to make things easier on the GM. But the decision is up to the GM.
Personally, i would rule that it happens at the end of the creature's turn that caused the effect.
Otherwise, the rules are undefined and you are subject to table variation, as this decision is put on your GM hands.