I should say being London based I have no personal experience of this. However, it does seem to me that the connections issue revolves around a few things like crowded and uncomfortable waiting areas, relatively long wait times, unfriendly staff and the lack of premium routes.

All of which could be solved with some limited investment in more buses, better control and passenger information systems, more and better trained staff and a better offer for premium passengers. Probably harder to identify will be a little more space for waiting, but some of this would be negated with a more frequent service.

I know probably easier said than done but there are real benefits of reorganising things. For example IRROPs could be much more smoothly handled for LHR-JFK passengers if AA/BA are co-located at T5B and T8.