You have two different checkpoints. First, entering the US you will have Immigration and Customs. I can't address this for you, as I am a US Citizen with Global Entry so my entry into the US is ultra-fast.

Second, when leaving the US you will have the normal TSA Checkpoint as is required of anyone flying in the US. The best way to get through TSA is with Pre-Check, however BA is not part of the TSA Pre-Check program.

There is a little-known arrival level TSA checkpoint in the connector between TBIT and Terminal 4 (AA) which I recommend. The "normal" TSA lines in the departure level of TBIT can be long. The arrival level TBIT-T4 connector TSA checkpoint rarely has any wait at all. I will give you very specific instructions, as you said you have never been through LAX (my home airport).

If you are checking bags: Get your ride to drop you at the TBIT Departure level and go to the check in counters. Check in your luggage there and get boarding passes. Resist the temptation to just follow the crowd to the TSA checkpoint and departure gates.

Instead: Turn back around and move back towards the glass doors and windows of the street. When you are close to the glass doors, turn right and go along the inside of the building parallel to the street in the direction of Terminal 4 (AA) until you see an elevator (="lift") on your left. Take that elevator down to the Arrivals area.

Exit the elevator and, staying inside the building, turn left and walk past a bunch of shops and restrooms. You are walking in the direction of Terminal 4 (AA). At the end of this hallway you will see a TSA checkpoint. Go though the checkpoint, then go up the escalator within the connector back to the TBIT Departures level. A right turn will take you to TBIT. A left turn will take you to Terminal 4 (AA), and access to T5 and T6.

For BA, you will turn right and walk along the connector corridor. Through a window you can look down and see the long TSA queue that you just avoided. Go on further to the lounges. The Qantas First lounge is the first lounge you will encounter as you exit the corridor and enter the terminal.

If you are not checking luggage, and already have your boarding pass, you can skip the entire Departures level trek. Ask your ride to drop you at the Arrivals level. Go in the glass doors, then turn to your left and go through the long hallway, with the restaurants shops and restrooms, where at the end you will find the TSA checkpoint.

Since I have Pre-Check and usually fly AA, I don't use that checkpoint personally, but people have told me that they were amazed at only waiting in a very short or non-existent line. This checkpoint was designed for people coming off international flights and connecting on with AA or other airlines in TBIT. They come out of International Arrivals, then go down the hallway, pass through TSA, then up into the connector that takes them either to TBIT or the AA terminals.

I hope you have a GREAT time in Los Angeles. My favorite place to take tourists is the Griffith Park Observatory and Planetarium. You get a beautiful view of the city and a great photo-op for the Hollywood hillside sign.