I have never been refused. I regulalrly (well did until July) transited Liverpool, Manchester or Belfast. Now and again Dublin or East Mids. And not once was I refused. I have had to hold outside the class D for a few minutes and had to accept vectors through the airspace.
I usualy give them an early call as possible. Liverpool have been the best for me even during peak Easyjet hours closely followed by Belfast. East Mids have given me the biggest run about as has Dublin