We've just finished a session with the Dogs dealing with some children gone very bad.

The last scene took place in the school house. One of the dogs was facing off 3 off the worst children, while another had led the majority of the children present into another room where to see if they'd be more cooperative outside the influence of the ring leader.

Dog A is badly injured and requires a healing conflict.

One of the children from the back room is about to see his sister has been knocked about and will immediately attack (possessed)

As far as I can tell, these are both follow on conflicts, both happening in the same place. Should they happen concurrently, or should 1 happen before the other?

As I see it, if Dog A wants to be in the conflict involving Dog B, his healing has to wait. However, what happens if he gets injured again?

I think group judgement is the best guideline here, do what the group feels good about and feels reasonable.  I know that may be a facile answer, but I think it follows the book's guidelines on such things.  Generally I'd go for doing the healing conflict first as a "quick patch" scene, to get the action flowing immediately into the next conflict.  But that might not be in line with the group sense of realism and the nature of the "bad injuries" (I think a lot depends on the nature of those - if he's "just" losing blood that can be resolved quite quickly, if he lost an arm that's a bigger deal).  There are possible other Conflicts here, too, such as "Do we get Dog A away from the child about to attack?"  If I were players, I might go for that, and so that would be the first Conflict, then the PCs except for Dog A face the attacking child, then the healing Conflict.  This would especially fit the scenario where Dog A needs more serious medical attention and mainly they just want him out of harm's way to get that care.