Can a neck injury affect the peripheral nerves and cause nerve tingling/numbness in upper extremities ?

Yes. The nerves serving the arms originate from the cervical spinal cord. Any injury to the nervous system should be taken seriously and you should consult a doctor to ascertain if consultation with a specialist may be needed. Wish you good health!

Depending on extent of trauma, might see nerve root compression secondary to ruptured disc or foraminal narrowing. But occasionally nearby brachial plexus is hurt instead. Takes a careful exam to sort out