Is it possible to recovery fully and quickly after a ganglion cyst removal while caring for a 16 month old?

Yes: Depends on the weight of your 16 month old. If they are walking, then let them do so with hand support. Wrist splint will help with initial discomfort. The splin can be found in most drug stores in the bandage aisle.

Protect with a cast: Yes. Typically recovery after an excision is 3-4 weeks. A good option is casting during the first three weeks (fingers free) which allows you to use the hand and minimizes the chance for recurrence. While in a cast you can fully use the hand, with minimal discomfort with the confidence that the surgical site is adequately protected.