Impossible to prevent - cables are very long, in deep water and cannot be patrolled along entire length hour by hour.
Locations of these cables are also clearly marked on many maritime maps (to warn trawlers not to damage them). Also locations where they come ashore are known.
Expect UK government to invest / encourage investment in more satellite bandwidth and also in radio base stations between France and the UK, surrounded by military grade security - both to provide basic web communications in emergency.
But international web traffic would still be so slow that it would be liable to gridlock, without the entire nation being asked to stop all non essential web use.