LHR isn't really the best airport for onward travel to Liverpool given the long hike from plane to tube and multiple changes between tube and mainline stations. A much better bet is to fly ATL-MAN. Immigration for the morning TATL arrivals is usually quieter than at LHR, plus it's an easy walk to the airport train station where you can get a direct service to Liverpool Lime Street (no change necessary). MAN to Liverpool is 1 hr by train (vs 3 hr from London).