Your modem gives the public IP address to the router, and the router assigns the private addresses to the devices connected to it, while the network switch doesn’t handle allocating IP addresses and serves as the extension of the limited ports on the router, to receive more devices. In this scenario, all your devices with private addresses are safe as they are not directly visible to the internet.
If you put the switch first, your internet provider does not offer multiple public IP addresses before the full transition from IPv4 to IPv6. So one or all connections will likely fail and all of the devices connected to the switch would be exposed to the internet. Once it talks to the first unit, everything else will likely fail. If the switch is first, then each port on the switch is a different IP address that doesn’t exist as those are LAN addresses. I am sure someone can configure a way to make it happen, but would be way overkill, etc. for a residential situation.
You can plug your BI computer directly into the router. But your switch for your cameras will not be connected to anything other than the cameras and the BI computer.