@Scottparker50
One reason why no list of serial numbers appears is if you are trying to install your Google Nest camera with the wrong app.
We have a Google Nest Battery Camera, model G3AL9; it gets installed in the Google Home app. Your post says you have a "Nest Cam (battery)" but says you are trying to install it using the Google Nest app. Use the Google Home app instead.
If you did not save the QR code sticker that was on the front of the camera (Google Nest's instructions say to save it with your Quick Start Guide), then you'll need to select "Continue without scanning" during the install, and will need to enter the 6-character alphanumeric Setup Code printed on the bottom of the camera. See the install video here: https://support.google.com/googlenest/answer/10919314.
Also, Google Nest does not make it easy to change Wi-Fi settings on a Google Nest camera; that's why their first recommendation is to set the network name (SSID) and password on your new router/modem to the same values used on your old router/modem (https://support.google.com/googlenest/answer/9223711).
If you can't do that, then you'll need to do a factory reset (not a restart) on your camera before trying to reinstall it (https://support.google.com/googlenest/answer/9252162?hl=en&sjid=8773053829550587528-NC#zippy=%2Cnest...).
