From your description you are travelling before LAX will be included in the scheme, but there is a trial running for connections from the USA which negates the need for a UK security check. If you're looking at a travel date several months hence, you might be lucky.

https://www.flyertalk.com/forum/brit...ity-check.html

I don't think it's been covered yet, but if you make an airside transfer you will not have to clear immigration or customs unless you're connecting to a UK (or Common Travel Area) destination, in which case you will clear immigration at Heathrow and customs at your final destination. Customs is only ever a walk-through affair anyway