I have a friend that has Renauld's (Not sure if this is the correct spelling or exactly the name, but if you Google or look here, you should be able to find it.) Disease where her fingers cannot be exposed to the cold because of severe pain. She can have very drastic and major consequences if she doesn't keep her fingers warm. I don't know if this could even be a thought as to what is happening to your son, but I did want to throw it out there for you.
Aw, bless his precious little heart! I have a 3 year old, and I know your heart just breaks for him. I'll be praying for him. I think it might be the nerve endings in the affected finger, since they were severed, and may be more susceptible to hypersensitivity to the cold. A friend of mine has a little girl who lost some fingers in a lawn mower accident, and she is having some of the same problems. Again, I'm praying for your little guy!
One more thing...I did some researching on Google by looking up "amputation" and "cold" and "finger" and cold intolerance in people who had lost a finger was mentioned a lot. Just FYI. :)
I'm so sorry your little one has to go thru this. I lost two fingers when i was 11 years old, & know exactly what he's going through. The only thing i ever found that helped was putting inside the gloves one of those heat things they sell hunters to put in their gloves to keep warm. I'm sorry i don't remember what they are called, but you can buy them at any sports store where they sell hunting supplies. I think they also make now gloves that are battery heated... I'll keep your little guy in my prayers for a while....