TLDR: What, if anything, are the implications for a child's (future) sins, if any, if they fall ill?
The rest of my question is a preamble and axioms for my assumptions.
In Islam, children are considered sinless. This is supported by many proofs, including the famous hadith:
The pen has been lifted from three: From the sleeper until he wakes up, from the minor until he grows up, and from the insane until he comes back to his senses or recovers. (Collected in Tirmidhi and Nasaa'i's books)
In addition, hardship in general is a test from Allah, and forgiveness for our sins:
I heard Allah's Messenger (may peace be upon him) as saying: There is nothing (in the form of trouble) that comes to a believer even if it is the pricking of a thorn that there is decreed for him by Allah good or his sins are obliterated. (Saheeh Muslim)
This is also supported by rasulullah's statement to a sick person:
The Prophet () visited a bedouin who was sick. Whenever he visited an ailing person, he would say, "La ba'sa, tahurun in sha' Allah [No harm, (it will be a) purification (from sins), if Allah wills]." (Saheeh Bukhari and Riyaad as-Saliheen)
Therefore, for adults, we know that illness necessitates forgiveness of your sins. That's great news.
But what about children? Being inherently sinless, there is nothing to forgive. Is there any recompense or reward for them in the hereafter? Perhaps forgiveness for their future sins?