M.K.
If you pick him up and he stops crying, I would first assume it's not a health issue. But when you mention the puking, perhaps start with that subject first to your SIL and suggest he see a doctor as it could be an indication of health problems that give the poor guy pain enough to cry a lot. You can gauge her reaction to that and then if necessary mention the crying - if she thinks he's crying because he's in pain she may see a Dr.
If the Dr. can't find anything wrong, it may be like my daughter. I personally know how trying this can be - my daughter did this. The best thing that we did for her was hold her in a sling as we moved around the house, doing dishes or cleaning or whatever. Some kids need more closeness than others. If that's the case with your nephew, he will grow out of it. Plus,he'll grow out of it sooner the more you hold him now - it will increase his sense of security until one day, he'll want to be on the floor playing like the other kids. Good luck to you - it sounds like you have your hands full.