I have a ganglion cyst on my right wrist, palm side. I am experiencing numbness and tingling in both hands overnight. Is that normal?

Nocturnal tingling. of the hands usually is due to carpal tunnel syndrome. It's not uncommon to see carpal tunnel in individuals who also have ganglion cycts. Have your neurologist do an EMG / NCS to clarify matters.
No. A ganglion cyst although uncomfortable and particularly 1 sided shouldn't do that. It may be caused by the nerves in your wrist and maybe something like carpal tunnel disease. See you doctor for assessment and management and I hope you get better soon. The ganglion cyst sometimes ruptures and resolves and sometimes needs removal.