That's a horrible situation to have experienced, for both of you, so it's not in the least surprising that you feel shaky and upset. Here are my thoughts, in no particular order of importance, because that's up to you:
Write to whoever is responsible for managing the swimming pool, describing exactly what happened. Enquire whether they have noted the incident in their accident log, and ask for a photocopy. Ask if they have carried out a Health & Safety investigation, and (if appropriate) have reported the incident to the Health & Safety Executive. Give them all the information you have about door hinge protectors. The hospital should have taken notes of where and how the accident took place, but I don't know what they do with that information - maybe when you're next there you could ask about that.
Young children do have a much greater physical ability to recover from traumatic injuries than adults do. Although you were told there was a risk of losing the finger, that doesn't necessarily mean that will be what happens. They have to tell you of the risks, because it would be grossly unfair to do otherwise. Most surgeons and nurses want to see their young patients discharged from their care with a complete recovery.
If the worst happens, yes, that's horrible to contemplate, but children are incredibly versatile and even if his finger can't be salvaged he will be all right despite his injury.
Time really is a variable thing, isn't it? When you're counting down the hours until Tuesday. Been there, and it's never easy. Talk until then, if you need to. And then let us know how it went.