May I suggest a simple solution for pilots involved in medical emergencies?
Make a 'Pan' call (perfectly legitimate), and say 'I require you to make a 999 call for a paramedic ambulance, confirm you will do this'. ATC will, I am sure, act as required.
I mention 'paramedic' ambulance, by the way, as I understand that there is a distinction, and that this is the appropriate thing to say if life is in danger, as opposed to, say, a broken leg or whatever. Any senior medics please correct me if I'm wrong, but this was told to me by a medic.
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'Brighten my Northern Sky' Nick Drake R.I.P.