I'm looking for any recommended exercises or training that my son can do whilst his right wrist recovered from a buckle fracture. Most core and shoulder exercises seem to rely on using the hands/wrists so can anyone recommend things he can do to keep the rest of him in shape (or improving) without compromising the wrist?
Sit ups or aerobics might be good, or him doing things 'plank' type exercise for his core where he uses his elbows and sticks on leg out behind him and then the other, but it possibly won't set him back too far in the big scheme of things if he rests for a bit. He's still got his lifetime ahead.
It's interesting how much one thing like a wrist can limit you.
As Timmd says you can certainly train core plenty without using the wrist. Elbow planks (front and side), supermans, a wide variety of crunch type exercise, hollow body, v ups, russian twists and lots more.
He could also use the opportunity to do leg training. Progressing pistol squats would be worthwhile for climbing I'm sure.
He could also train his good side are in a variety of ways - any exercise with a dumbell for starters. Stuff on a bar or fingerboard could be done with pulleys and/or resistance bands (unless he's already super strong of course). Or if access to a gym he could do one arm stuff with a cable machine too. Obviously not ideal but I've heard it said that the body will try to maintain some symmetry so training one side will result is less strength loss on the other while it's inactive.
Another thing he could work on is flexibilty. Lots of yoga stuff on Youtube and elsewhere.
One handed climbing - this is my response to various finger and hand injuries, and it keeps me in shape and stops me getting too out of practice.
It helps you to use alternative moves, find different ways of using a hold, slows you down and improves your balance and footwork. One useful move is a semi dynamic move up to the next handhold, such that at the top of your (slow) spring up when you stop rising before you come down again you are briefly motionless and can just reach across to the next handhold. Elbow hooks are sometimes useful, chin hooks are best avoided!
All this works much better while top roping, and while climbing lower grades than before. Lead climbing is harder, you always have to clip with the bad hand and be relaxed and ready to let go and fall at any moment (no desperate grabs !).
Bouldering is harder still as falling onto the injured part is a no-no, and down climbing one handed is a challenge!!
Finally, swimming is very good exercise. I once broke my leg just before a skiing holiday (gutted !!) so my parents still took me to Italy. I spent the week in the pool looking out the windows at the mountains, and got a lot strength back in my weak jelly like leg. I even made it (determinedly) out onto the slopes one day hopping with a pair of ski sticks...