It is likely to have gone down without causing any obstruction, Kindly observe if you have any pain, irritation of the throat, any breathing difficulty or consistent coughing, in which case you should visit the ENT so that you can be examined to look out for any obstruction and have it removed accordingly. If the problem persists or worsens visit the ER at the earliest.

You need not worry if you don't have significant problems such as severe throat pain, severe pricking sensations while swallowing. Usually piece of tooth pick will easily pass through our digestive system.