PHB pg 268 said:Ongoing Damage:
Some powers deal extra damage on consecutive turns after the initial attack.
So, to the OP, its option B.
Since the Fighter's power damage the creature on the creature's turn, they would not take the damage until their next turn. The creature would get a chance to save at the end of their turn however.
If the creature did not make their save on the first turn and they were still adjacent to the Fighter on the next turn, then the Fighter would deal the 1W and the 10 ongoing.