No horrors.
We flew thru LHR connecting from BA to AA on the way to JFK. Made the transfer within 20 minutes (however we were in Club Europe on BA and could use the FastTrack).

On the was back we transferred from T3 to T1 in 15 minutes (we were the first flight to land into Heathrow that day).

Our luggage was checked-in in L.A. thru JFK thru LHR to KBP - all three bags were transferred to BA and arrived with us.