I think my child has nursemaid's elbow. Should I take him to the er?

Your doc or UC is ok. Nursemaid's elbow is a partial dislocation of the radial head from the joint socket. It is easily reducible by a trained medical professional, however the er may not be necessary. If it's after hours and you have a local urgent care nearby that sees children, that's a good option. During the day, your pediatrician is the best person to take care of it and it only takes a few seconds to treat!
Nursemaid elbow. Definitely, you need to take him. It has to be reduced because it can cause swelling and more pain.
Take to pediatrician. Nursemaid elbow is usual a result from pulling on the child's elbow. This can result in exquisit pain and the child usual holds the arm flexed at 90 degrees and cradled in the other hand. Though the problem is a simple dislodging of the bone that is stuck on a ligament, it's a good idea to have your child's pediatrician confirm that there's no concern for an underlying fracture (broken bone).