Stu H,
I have been at the college since Sept. The medical has no bearing on whether, or when, they give you a start date; many people do not have their medical (CAA CL1) until they have already started college.
With regards to your notice period at work, I fell upon the same problems; which resulted with me handing in my resignation on the basis that I had passed the medical. There were a good few of us who resigned without the slightest whiff of an employment contract.
I can't offer any more insight than that I'm afraid, apart from to try HR again and explain your problem.
See you around.