Would a pinched sciatic nerve cause pain in wrists and numbness/tingling in hands or feet?

Feet maybe, hands no. The sciatic nerve runs from the gluteal region (your rear end) down the back of your leg, then branches to innervate the back of the thigh and leg as well as the outer part of the lower leg and then parts of the feet. An injury may cause those symptoms there, as well as problems with motion. The major nerves that serve the hands are the radial, ulnar and median.
No. A pinched sciatic nerve will not cause wrist symptoms or numbness in the hands. A pinched sciatic nerve may cause symptoms of the leg.

Related Questions

I think I have damaged my sciatic nerve, I don't know how but I have random tingling and numbness in my feet, legs, arms and hands. Why could this be?

See your doctor. The sciatic nerve is actually a bundle of nerves that starts in the low back and supplies the lower extremities only. Numbness and tingling of both feet, arms, and hands is not consistent with a sciatic condition. See your family doctor or internist to find the cause of your symptoms. Read more...
See below. Damage to your sciatic nerve would not dause arm or hand problems. Read more...