If wifi that my Nest camera is connected to goes down, can it reconnect to an alternative wifi? In other words if the "thieves" cut the broadband cable, but I still have a 4G router as a backup, would it work?
A better approach is to get a broadband router that will fall back to using a 3G/4G USB stick if the broadband line goes down.
This means any devices (such as your Nest cam) doesn't need to know about 2 different networks, the router handles all that for all devices.
There are plenty of these on the market as it's a standard fall back for small businesses that don't want to install a back up second line.