Preface: this is going to be expensive and likely not valuable to anyone else, since it is non-standard.
First, I would do away with the Coax part of your idea. You're talking about encoding multiple channels of HD video onto a single coax line. Cable and satellite companies do this with expensive equipment with compression. HDMI has literally 20 pins, you can't just do this simply and electrically. HDMI over Cat5e has some compression and might be useful to you, especially with my next suggestion:
If you're not putting all the signal on one cable, how do you do it? Multiple cables. Modern TVs have ~3 HDMI inputs and switching between them is simple like changing channels. If you're talking about 3 different video signals and 3 TVs, you're looking at 9 cables crosslinked like this:
Obviously, the more TVs and the more video source "channels" you have, the more complicated this will be. This also requires HDMI splitters and multiple cables run from your sources to your TVs. On the upside, you can run this with off-the-shelf hardware and full HDMI resolution.
Controlling your laptops could be as straightforward as a remote software on a tablet near each TV that you use instead of/in addition to a TV remote.
Having said all this, the proper and much cheaper way to do this is using the built-in "smart TV" software on each television, and if that's obsolete or not present (older televisions) to upgrade with a "streaming stick" device such as Chromecast, Fire TV, Apple TV, Roku, or other. For around $50 per TV, these will tune into any number of online streams and save you the cost of laptops, wire runs, adapters, et al. You can control the streaming of most of these from a smartphone or tablet (and the same smartphone/tablet can control whichever TV you happen to be at) making this solution superior in many ways to your proposal.