Short answer. Yes, in principle. Where support is given by a stranger without the knowledge of the person obliged to give it, the giver may claim it from him. The exception is where it appears you gave it without any intention of being reimbursed.
What the law says
When, without the knowledge of the person obliged to give support, it is given by a stranger, the latter shall have a right to claim the same from the former, unless it appears that he gave it without intention of being reimbursed.
Family Code, Article 206 — Reimbursing a Stranger Who Gave Support. Read the full provision →
Stranger means stranger to the obligation
You are an uncle, so the word may read oddly, but it is used in a technical sense: a person who was not the one legally bound to provide the support. The obligation to support the child rests on the father, and you had no duty to discharge it. That is what makes you a stranger for the purposes of this article and what gives you the claim. Someone who was himself obliged to support the child is in a different position entirely, since he was paying his own debt rather than another's.
Why the lack of knowledge matters
The article addresses precisely the case where there was no chance to ask. Support is often urgent — a hospital admission does not wait for the obligor to be located and consulted — and the law does not want a person to hesitate at the ward door for fear of never seeing the money again. So payment made without the knowledge of the person obliged is recoverable rather than treated as officious meddling. The claim runs against the person who was bound, and it is measured by what was actually given.
The exception is the real battleground
Recovery fails where it appears that he gave it without intention of being reimbursed. Between relatives that is exactly what the other side will argue, and family circumstances often make it plausible: an uncle who helps a nephew is usually helping, not lending. Notice the wording, though — what matters is how the intention appears from the circumstances, not what the payer privately felt. A payment made with nothing said either way, in a family that habitually helps one another without accounting, is a difficult claim.
Say so at the time, in writing
The way to keep the claim alive costs nothing at the moment of paying: a message to the father telling him what you paid and that you expect to be repaid, sent when you pay rather than months later. Keep the hospital bills and official receipts in your own name, and the proof of the transfer. And be clear about what you are claiming — the support actually given, not the inconvenience — because the article gives you back the thing you supplied, not damages for having had to supply it.