Just recharge as u do.
Nothing will deliver to your address.
Earlier day after opting for Prime they used to send a thanks letter and a key ring . But now they have stopped that. Just recharge as u do.
When the amount has been debited from your account sometime it might take around an hour to get the services to run. Check with Customer service if you still do not get the service activated.
It will show delivered to your address, ignore it, even if you do mobile recharge or any bill payment then also it will say the same. Please try to login into Prime Video with your email and password, you should be able to login there, with same email address and password you should be able to login in app also.