They can play the heal at any point in the turn.
The deal here is that the heal acts as an interrupt instead of sorts for the discard. Once the heal is played it is as if the character was never in the discard pile, no matter when the discard occurred or when the heal is played. That is why healed characters retain their weapons, experience bonuses, etc. (or that is, at any rate, how it has been explained to me.)
(BTW, I always DD the boated ones during the discard phase. That way DD prevents you from playing a heal.)