If your test is on Saturday the 20th, then you will still officially be at the address on the licence on the day you pass. You don't need to do anything.
If you pass, your new licence will be sent to your current address, then when you move at the end of the month, you must notify DVLA (online) of your new address. A replacement licence will be sent to you free of charge at the new address.
In all cases with house moves, it is recommended that you set up a royal mail redirection for a short time so that even if you've informed most companies of your address change, organisations that you've forgotten about can still reach you.
For those who have already moved house, you should have already notified DVLA before your test, so that you turn up with the new address on your licence. Address changes on test are no longer possible. If you turn up for test having moved house and not informed DVLA, then your licence will be returned to you and if you pass, must send off a D1 form with your licence and pass certificate to get a full licence sent to your new address.