You should see the plane in your fleet as soon as you complete the lease agreement. There's no waiting period.
You're probably not completing the lease, though. After you click the "Lease" link you're taken to a screen where you can see more details for the plane. There, you have to press the button labeled "Lease aircraft". Are you doing it this way?
If you're sure you confirmed the contracts, go to Operation department and see if the money was deduced from your account.