As I see it there are several options:

Assuming coaxial video cable has been run....

1) Assuming NO HDMI on display: Use HD Fury converter at the source end to convert HDMI output to YUV and keep existing cabling to screen as-is. We have done it often (we used to sell a lot of Fujitsu Plasmas!) and it works fine.

2) Assuming there is an HDMI input available on the screen: Use an HDMI to coaxial cable converter TX-RX pair. Wyrestorm do one, and although I have not used it, I have seen the demo and it works.

If YUV has been converted to Cat5 and Cat5e cable has been run, or there is a spare one there...

3) Assuming there is an HDMI input available on the screen:Use an HDBaseT Transceiver pair to send HDMI to the screen over Cat5. You will of course need to do a test first to ensure the Cat5 installed is of a type, and has been installed correctly, that it will support HD video