Most airlines have a time limit for connecting bags, if you exceed it, then you must collect your bags at the connection point and recheck the next day. The time limit varies with the airline. I recall a trip to Sitka, AK, in which we flew SAN-SEA in the evening, then continued SEA-SIT early the next morning. We exceeded the eight hour rule for Alaska Airlines, hence our bags were checked only as far as SEA, where we collected them and rechecked them the following day. We already had boarding passes for the second flight.

As others have correctly posted:

1. You must clear immigration and customs in the first port of entry, i.e. LAX in your case.

2. Following customs, but before the lobby (i.e. still inside the restricted international arrivals area), there is a "baggage recheck desk" where you give the airline back your checked bags. The tags show both flights on normal/brief connections, and you then proceed to the lobby, go through security again, and then go to your connecting gate (note: since you have had access to your checked bags during customs, there is no "sterile" or "airside" connection to your gate for the second flight; you have no choice but to continue to the lobby and go through security again). The process at the baggage recheck desk is quick and informal. Since your bag is already tagged with the connecting flight and destination, they simply take your bag off your cart and place in on a belt. They may ask to see identification and/or a boarding pass, but the process is very quick and efficient. In some US airports, there is one recheck desk for all airlines. ORD re-configured their desk into I believe three desks, one for each major alliance. On a recent trip ARN-ORD-SNA, the SAS desk at ARN was unable to generate a boarding pass for the connecting ORD-SNA flight on United, so I had to request a pass from the recheck desk at the same time I gave them my checked bags. In some cases, you simply remove your cabin bag from your cart and the recheck people take your luggage cart and place all of your bags on the belt for you, and you simply walk out to the lobby.The only time I've ever waited more than a few minutes for recheck was once at EWR (Continental) where there was a ten minute queue, I must have arrived at a peak arrival time.

3. There is no "requirement" to hand over your bags at the connections recheck desk. If you wish, you may bring your bags through the international arrivals door into the lobby, but you'd have to check them again.

Check with your airline re: the maximum allowed time for connections (i.e. the maximum time where you can check the bags all the way through). If your connection exceeds this limit, at LHR the tags will only be checked as far as LAX (the tags will show only one flight and one destination). That is what happened the time we laid over at SEA: the tags said SEA as final destination. I don't know of any US airlines that allow check-in of bags the night before a flight (common in Germany and some other European countries), so if the tags say LAX as your final destination, bring your bags with you to the hotel and check them back in the next day. Remember, you can use curbside check-in if you wish, since LAX-HNL is domestic. The service is free but most folks tip the skycaps $1-2 per piece of luggage. Quite often, it's faster to use the sidewalk skycaps than to wait in line inside, particularly since you will already be checked in and probably will already have your LAX-HNL boarding pass (issued in London). You can't use the skycap service for international journeys, but you can do so for domestic.

4. Re: the flight home. If your 12 hour layover in LA exceeds AA's baggage connection time maximum, then your bag will be checked through only as far as LAX, where you will reclaim it (domestic baggage claim, no formalities) and then recheck it at LAX in the afternoon. You cannot use the sidewalk check in for an international flight. Be advised that the minimum suggested check in time for LAX Economy is three hours. You could reduce it to two hours if you are in Business or First class. Even if you have your LAX-LHR boarding pass and are checked in via kiosk or online (or at HNL), you still have to wait in line to check your bags at the counter, since you can't use the skycaps.