I agree with SwordGuy. The mortgage company notifies register of deeds of the payoff and the rest follows from there. A month is about the typical time you should get notice.
Congrats on paying it off. I see you got a few of the disclaimers about investing being a better approach. But here are two key points that most people don't mention (they apply to me too, as I just paid off my mortgage). First, I eliminated a $2000/month payment by paying off my house, and THAT is what enabled me to retire early. It is about cash flow. I would need an extra $600,000 in investments just to cover the mortgage. Now without the mortgage, I can retire on my current investments. So I actually retire earlier (not as much wealth accumulation, but better cash flow). Second, a paid off house offers flexibility, you can still use the equity by HELOC or refinance to buy rental properties if you want. So it provides flexibility and options--or just peace of mind.
Good for you-all the best.