[moltuae]

Most 'dumb' devices will reconnect without issue to a different WiFi infrastructure if it uses a 'known' SSID and password, although the subnet and gateway addresses may need to be the same in some cases (especially if you have any devices with static IPs). However, some devices (such as Windows-based PCs) will detect that the router's MAC and/or NetBIOS are different, which will default the network type to public/untrusted. But you can easily switch it back to private of course.

ISP-supplied modem/routers are terrible and the built-in WiFi capabilities of most routers are quite poor/limited. Also, just like using an ISP-based email account, using and configuring an ISP's router is a bad idea because it puts reliance on the ISP and makes switching to another ISP more difficult.

I don't do residential IT, but for my smaller business customers (and my own/home setups), any ISP-supplied router goes straight in the bin. I instead obtain the broadband connection details (inc. any login credentials) and connect a business-grade firewall/router directly.

For WiFi, I always disable any router WiFi features and install Access Points instead (usually Ubiquiti/UniFi units), which makes extending a WiFi network simple. Access Points also provide a far superior WiFi service and are easy to replace or upgrade when newer/faster models become available. For smaller setups, I usually use DrayTek routers. Ubiquiti also make some very good routers (and an all-in-one product called a 'Dream Machine', which incorporates a router and an Access Point controller) but I'm yet to use one (mainly due to their current limitations in handling multiple public IP addresses). Some of the DrayTek routers support mobile broadband (LTE) directly. Some models have a SIM slot and LTE antennae, while some cheaper models use compatible mobile broadband 'dongles' instead.

Something to bear in mind with most mobile broadband providers is that they usually don't provide you with a public IP address. Instead you'll receive a pre-NAT'd connection with a private IP address (I think Three tend to give you a private IP in the 10.0.0.0/24 range). You need to speak to them if you need a public IP address (which you will need if you need to forward ports for cameras, etc).