Ok drauk ,
Big skies we are talking London here they don't exist over London big skies they are full.
There are a lot of aircraft flying in a small amount of airspace with minimum spacing between them. An aircraft responds to a TA and he climbs /descends into other aircraft causing chaos.
The frequency you are operating on is not the same frequency that the other aircraft are on , the controllers are not even in the same building. You declare a Mayday the controller has numerous tasks to deal with a phone call to LTMA is going to add considerably to his workload and by time phone call as got through it could be to late.
I have sadly in my career had to deal with numerous Maydays , some have become fatal accidents. In most of those which the aircraft has crashed (all light aircraft) they havent even had time to issue a Mayday.
I am not telling you how to fly your aircraft I am advising that a climb into the LTMA has very serious consequences not just to yourself .