Honestly Wexflyer, the food in the FL's leave a lot to be desired. The LAX club in the early am usually has on offer some Danish pastries, bagels, croissants and breakfast cereals. Sometimes they have some lox but they tend to hide it away in the "upper refrigerators" to the right of the service area where they keep the soft drinks, etc. You will have a selection of fresh fruit which IMHO always needs a few more days to ripen. At that time, since Qantas is not footing the bill the champagne will be Korbel!!. They have some premium spirits on offer and usually a few chocolates. In short, not much and not very good. The evening hours when the Qantas flights take off have a much better selection of higher quality items.
As far as the LHR FL there is a very good reason to go to the CX lounge. I know you did not want to hear it but facts are facts and the gang her at FT know what they are talking about. The nonexistent food in the LHR FL sucks! If you are on the LHR-LAX nonstop you really will be left out in the open. In the early morning they set out the Danish and yoghurts etc. and from what I have experienced when they are out for the day they are out. They don't put out the little finger sandwiches until 12 noon after the LAX flight has departed. Last month when I took the non stop I arrived in the FL about 9:45 am and the only thing left was cereal and apples. Thats it folks. I inquired if there was anything else and the lounge attendant said she had finger sandwiches but was on "strict orders" to place them out no earlier than 12 noon. I Then went over to the CX lounge and was admited via my Platinum status and my F boarding pass. I was allowed into the F side (to the right) where there were fresh dumplings, all sorts of breakfast pastries, finger sandwiches and a small menu that you could order a caesar salad, Chinese noodles and some other items. I had the caesar salad and the dim sum along with a nice glass of champagne. It is not the wing or the pier, but it is far better than the AA FL.
If your status allows by all means bypass the LHR FL and go directly to the CX lounge. If you can't and have to go to the FL, I suggest a stop at one of the food stands in the chaos that is terminal 3. At least you will have something to eat.
I know this is not great news but this is how it is. I hope you have a pleasant journey.