Most A/V installers would have installed a wall outlet with the HDMI port built in, to make it easier to replace/repair in in the event that the cable or plugs failed for some reason. If it is just connected to outlet, you should be able to just unscrew the outlet/wall plate and fish a new HDMI cable down the wall, that you've tested will work with the Apple TV and Television.


https://www.homedepot.com/p/Commercial-Electric-1-Gang-HDMI-Wall-Plate-White-180904/300412562