The Sea King heading north could well have been going to land in Regents Park to transfer a paitent for Great Ormond Street Hospital.
I've seen a number of helos landing there, RAF Sea Kings, Bristows Sea Kings, AA A109s with a GOS Ambulance in attendance...preferred landing spot is right at the north end, east side by the zoo. The grass area is easy to clear of users and access from the Outer Circle is easy.
As for the King Air I have seen it as well, a road watch type mission, with Capital Radio?
Just a thought.