If the reason they want to get hold of a legal next of kin is to get consent to a particular procedure, being able to get hold of someone else is of no help whatsoever to the medical staff.

AFAIK, in the UK nobody else can consent to medical treatment for an adult. If someone's incapable of giving consent doctors make the decisions. They will discuss options with family/close friends, but there's not a consent issue. The 'legal next of kin' thing is a bit of a myth in this regard.