Actually if you arrive at 5:55 am then I think you are relatively early in the rush. I have only done T3 to T3 connections (to SAS) in recent memory. When I arrived early like that it took me about 20 minutes to get through in October. But I noticed the line growing rapidly behind me as I waited there. Apart from that I have boycotted LHR because of the bag issue since August.

I would be concerned about bags getting lost/ delayed in LHR. It would be my last choice as transit airport for bags in Europe.