There are so many issues here.
For the people who are downplaying this by saying that people live for years on dialysis, while that is true, it is also a fact that people die on dialysis because it’s not sustainable, and it doesn’t work well for everyone. Dialysis only maintains stability, not health. A dialysis patient doesn’t go in for dialysis and come home and live a totally normal life. And if this child is in hospital, then clearly dialysis isn’t working as well for him as it would be hoped, and a transplant needs to happen sooner rather than later.
Being prep’ed for a transplant and it not happening is common, it’s called a false alarm. Assuming it was a deceased donor, the donor is identified, and then the tests are carried out to see whether the organs are viable, and sadly sometimes they’re not. I had two false alarms before I had my heart transplant.
With regards to the mother not being tested, there could be so many reasons why she hasn’t.
If the uncles have been tested and found not to be a match, is this because of a genetic condition?
Does the mother potentially carry the gene which has caused the illness? And if so, does she potentially have other children with the same gene and/or illness who could need a kidney as well in future, and therefore she’s not in a position to feel that she can give one to one and not the rest?
Or alternatively is she a single parent? And so if anything happens to her because of surgery what happens to her other children?
People are very simplistically labelling this woman as a bitch for not being tested, while sympathising with the OP for potentially having to look after a baby. Something which people have been doing for centuries.
And what about the bigger picture here? The OP has two children. These children are this boy’s siblings. So what are they going to tell them in the future? “You had a brother who died because mum wouldn’t let dad donate him a kidney”? Because that’s what it boils down to.
This child is nothing to do with the OP. Whether the father knew about him or not is entirely irrelevant. He may not have known about him then, but he does now. And if he wants to be tested to see if he’s a match then he, and only he, has any say in that decision.