You will not be double billed. The situation is as follows.
At the moment you have moved into a new property which is currently being billed by npower.
You correctly gave them your meter readings on the day you moved in.
You have also applied to have the energy taken over by another supplier.
There is a 2 week cooling off period required by law then your new supplier will apply to npower to take over the supply from them.
This may take in total 4/5 weeks altogether.
On changeover day you will be required to give new readings to your new supplier who will then bill you for what you use from that time.
Hope this is clear.
chris