Originally Posted by TWA884

Welcome to FlyerTalk NiHo!
Assuming that your flight to HNL is on American and your bags are checked through, you should be fine. Should your flight from LHR arrive late, you will be met at the exit from the plane by an agent who will provide you with a bright orange oneworld "Express Connection" card or ticket jacket. That will give you priority clearing immigration and customs inspections, dropping your bags off at the transfer desks at the exit from the Federal Inspection Services Facility and going through the TSA security checkpoint. AA flights to HNL generally depart from T5, approximately a 6-10 minutes walk from TBIT. Sometimes those depart from T4 (3-5 minutes walk) or TBIT.