I’m RN and I do a lot of rehab work. (spinal cord, cardiac, orthopedic, amputations)
I can’t comment on riding but I can comment on the healing of people with BKAs.
Like with injured horses, tell him not to rush it. 
I don’t know how the residual limb is closed (sutures? staples?), but he should be checking it daily. Don’t leave that only up to the MDs/nursing staff. When they do a dressing change he should look at it. If he can’t see it, they make special mirrors so that patients can see their skin (often given to patients with spinal cord injuries). The rehabilitation facility should have them. He needs to tell somebody immediately if it changes, more drainage, boggy to touch, whatever, speak up.
I’m assuming that the reason he lost part of his limb was due to poor vascular circulation status post his cardiac issues? If this is the case just keep an extra careful eye of how the limb is healing.
Healing properly ensures the best fit for the future prosthesis.
As a fellow eventer, I’m assuming your son is the brave type. It is fine for him to push himself but he needs to me mindful of that residual limb. If he goes “splat” which can sometimes happen in rehab, tell him to try avoid hitting that limb.
The prosthesis will also fit best if he waits until his limb is “healed”. Again try not to rush it. After he gets a prosthesis he should be very mindful of how it fits especially in the beginning. He does not want an ill fitting prosthesis to cause further problems with the limb. Examine for blisters, swelling, redness etc.
If you have any questions let me know and I will see if I can answer them for you.
