I'm not Now TV, just a customer like yourself, but I have swapped out a faulty white box for a new black box and also swapped out that black box, passed it to my son, and installed a Roku 3, all done without any problems whatsoever.
OK, I don't know if you have installed the new box with your old boxes details, if you have not done so then this is the step you are missing.
Also, remember that when you setup the new box you have to put in your wi-fi/network password as well.
However, if you have done these steps already, and your new box still does not work, then you should contact help chat on the following URL.
Contact live chat on the link page below, explain to them what's happened and ask them if they can reset all your devices (don't need the Serial Numbers of your devices).
Then start watching on your devices that you want to use where they will be automatically added to your account under My Account > Manage Devices (up to a maximum of four devices allowed).
Here's a link page below for live chat.