Unless it's changed the SAR heli at Lee on Solent never carried NHS personnel.
I spent a day with them several years ago and the winchmen had trained to the NHS paramedic standard, at a local NHS hospital and in their own time if I recall correctly. I m not sure what has happened since paramedics have become state registered though as many people outside the NHS have had difficulties in getting their registration approved.
Sussex Ambulance had an arrangement where some staff were 'trained' so that they could be flown offshore should the need arise. eg multi-casualty incident on board ship. I think like many similar ideas when managers moved it all went out the window.
I also approached Bristows for winch job but was basically told no chance unless you are ex-mil.
Also with the current pressures on the NHS ambulance services, especially Hampshire who like many are having trouble meeting the required standard I doubt that they would be able to consistantly supply a paramedic on demand to arrive within the scramble time, (about 5 mins from call in daytime from what I recall).