not wanting to run extra cabling, you could try HDMI to Ethernet adapters, one on each end of each ethernet cable, to connect your TV Hdmi ports to a HDMI splitter which in turn would be connected to your youview box at the cabinet in the garage.
Unfortunately that would be the trade off.
I wouldn't want to do without the ethernet ports, however I also have two satellite sockets and a coax radio socket in the rooms with the TVs. The other end of these sockets also come from the Televes Nevoswitch in the garage. Assuming that I can actually get hold of a coax to HDMI adapter, would such a cable support an HDMI signal to the TVs?
It is possible to get HDMI to COAX to HDMI adapters, they would be powered units and would convert the signal for transmission along the coax and back again at the TV end. I'm not sure how good they are or how much picture quality is lost in the conversion to COAX and back again.