Dom,
Just to clarify the Dispatch side of things - in the UK the term "Dispatcher" refers to what elsewhere is known as a Turnaround Coordinator, Coordinator, Ramp Duty Manager, and probably a few other terms.
// In other parts of the world (most others it seems) Dispatcher means Aircraft or Flight Dispatcher and can be a licensed position. I've worked in two charter airlines where we had licensed dispatchers (One in the US, and one in the Middle East).
The Flight Dispatcher is responsible for the flight planning, performance, and weather analysis for the flight, and usually flight following as well.
However, even where this is a licensed position, I've never known a medical to be part of that requirement. A company may require a company medical, but the license itself (to the best of my knowledge, and I have been responsible for those staff) does not require a medical examination.
Hope this helps a little
Brgds