Upon being asked why she didn't bring her child to the hospital back then she replies:
今から治療して間に合うものでしょうか！？
My understanding from context is that she's saying time passed since the incident and therefore it won't be possible to obtain a treatment? However 間に合う here is not negative. So I assume she's being rhetoric?
His answer being essentially "It's not a psychical wound, it's psychosomatic, therefore you will be in time."
There is no negation in this question, and it's not a rhetoric question, either. It's a simple question about whether they can make it in time for treatment. This もの adds a mild "generalizing" tone, as described in the following questions:
Therefore:
今から治療して間に合う(もの)でしょうか。
(Is it generally the case that) you can start treatment now and still make it in time?
The doctor responded with いいえ immediately after this, but his "no" was directed at her way of thinking up to that point, not at the question of whether he could make it in time.
