What happened was probably more like this:-
Hospital board meeting
(Director/Trustee) "Mr Chairman, do we have a designated spokesman to deal with press enquiries regarding the prime minister?"
(Chair) "Chief Executive, could you clarify whether someone has been appointed to deal with this?"
(CEO) "Yes, Chair, Mr Bloggins, our senior press officer is liasing with the Downing Street press team and will make such announcements are needed as regards recovery or worsening. In the unlikely event of his death, Mr Bloggins will also ensure that the Prime Minister's family is sensitively informed first and liase with the press office at Buckingham Palace and the Deputy Prime Minister before making any public announcement. Would you like him to notify you before any announcements?"
(Chair) "No, that's not necessary, but I'd like to be kept briefed of any changes. Are there any questions? No? Next item on the agenda..."