Well, think of the system in four parts. First, you have the satellite receivers and their associated hardware. Then, you have the modem(s). Next comes the routers, repeaters, "hot spots", and so on. Finally, you have the user devices that connect to the network.
The first part of the system will definitely change because the new service uses a different class of satellite than the old system and the old receivers almost certainly can't track MEO satellites. We've already seen evidence of this change with some new golf balls on the ships. The second part of the system (the modems) will probably also change to better interface with the new receivers and to support new bandwidth requirements and higher throughput.
But, once the signals get through the modems, they're just normal Ethernet signals. So, existing routers and WiFi infrastructure should work just fine. As I said before,
DCL might be upgrading this stuff, too, if the old stuff can't support high enough data rates to take advantage of the new satellite system. But they'll probably be doing these upgrades slowly, over time. Of course, the user devices won't change at all.
Whether the existing routers, etc. can work with inputs from two satellite systems isn't known. As I said, the outputs from the modem should just be normal Ethernet packets. Some routers (even some general consumer routers!) can handle two links and smoosh them into a single network. I don't know what Disney is using.
As I said, though, there isn't much benefit to keeping both systems active. Better to just use the new system and work with the vendor to solve any problems. I'm sure that they tested it for basic functionality before deploying it on the ships!