It seems that you don’t really want to connect but start an ex-third country itinerary to minimise the cost. There is a huge thread on ‘back to backs and quick turnarounds’ which compares the situation of almost all European airports. Obviously no back to back for you as you can’t be 100% certain you’ll be able to short check your luggage from London to Australia but you can see which airports are good and bad to turn around at!