Ok, so part one, like everyone is suggesting is you can probably make this fully automatic. I say, you don't even need to change the router IPs, but it's more difficult if you don't (you might need to setup 1:1 NAT to get sensible addresses and go from there). A new router with 4 interfaces, one to each modem and one to your network switch (if you can change modem IPs and disable DHCP on them, you can do everything with one NIC and a single collision domain, although you may not want to). You could get two devices each with 4 interfaces and use pfsync and CARP to make failover nearly seamless. There's appliances for this, but I've setup something similar from scratch on FreeBSD, use DSL if it's up, fail over to LTE otherwise.

Part two: what you asked for exists. Look for bypass NICs. There's a couple different flavors, but the commonly available pattern is a dual-port NIC, which you run a utility to switch from bypass mode (the two ports are connected via relays, the NICs are disconnected) to regular mode (each port serves one NIC).

If addressing is compatible, you could have three bypass nics, plug one end of each to your switch, the other end to a modem. The selected modem would be in bypass mode, and unselected modems would be in regular mode. (You could also do this with software bridging, only bridge the selected modem). I've had success with silicom bypass NICs, for older models like PEG4BPI, the -SD model is going to be easier to use PEG4BPI-SD has PCI ids set so that standard OS nic drivers will load automatically for the nics, but PEG4BPI has PCI ids set up differently and drivers won't start unless you modify them or can force them. You can actually modify the eeprom and turn a regular into a -SD, with just one bit flip, but that's an ordeal.