Can wrist pain radiate up towards elbow?

Yes: It could be related to tendons, bones, ligaments, nerves, or vessels. Sometimes overuse or injury.

Yes and it is a : Common occurrence. Especially nerve related wrist pain form carpal tunnel comporession which classically radiates to but not past the elbow. Any wrist structure that receives nerve supply ( which is any and everything) gets that neve input from the neck so the nerves pass by the elbow and pain can radiate upwards especially referred to other small sensory branches.