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When space marines rally from ATSKNF do they do it immediately after falling back, or during the start of that players next turn?

Niether. They take the test to rally whenever they 'activate' just before they move (page 46 in the BRB). Remember that they are still subject to the enemies within 6" criteria.

Same goes for all units, there is nothing in the rulebook that says a unit must test to regroup at the start of the turn.

Remember the ATSKNF rule states that they automatically pass tests to regroup.

So when it is their turn to run, if they can make a test to regroup, they will auto pass that test.