I'm taking a stab in the dark here, but you might be able to cut the ends off the coaxial cable and replace them with BNC, which might make this effectively SDI. You then get an SDI to HDMI converter.
I don't understand coaxial very well, but I know that "SDI is transmitted over 75 Ohm coax cable with BNC connectors." (Source) To me, they've always seemed like the same thing with different connectors. However, the actual wire in a coaxial might not be the same as the SDI spec requires. Often, wires have their specs printed on the pvc casing. See if it matches SDI specs, then you can try this with decent chances it will work.
But in your situation, replacing with BNC connectors and trying an SDI-HDMI converter should only be about $50. That's mostly for the crimp tool and a bag of BNC connectors. You can return the SDI to HDMI converter if it doesn't work.
I did run across this on a similar question: Edision HDMI Modulator Full HD Distribution over Coax