Most likely the second one was the original from wayback. The cable will
very likely go to the loft. Have a look behind the socket, see whether
the cable goes up down or sideways. Irrespective of whether there's an
aerial at the end of it, it could be useful if the resident wants to put
a TV set or FM/DAB radio there. A cable to the loft is very helpful for
connecting an outdoor aerial, especially if the socket isn't on an
outside wall.
After switchover loft aerials are more likely to work, not less likely.
If the cable is pre-1970 it could be the very old type of co-ax which
isn't really any good for modern use. Even if the cable is newer it
might be affected by damp, or it might be damaged in the wall.
If it appears to be very old thin cable, but there is an actual duct
(pipe) from the backbox going upwards (typical of 1950/60s upmarket
houses), remove the socket and replace with a blank wallplate. Possibly
label this 'duct to loft'. Leave the old cable in as a draw wire.
Bill
Bill