If you get the 6.30am Hilton bus, you could be at checkin by 6.45, almost certainly 6.50. If travelling business, the fast track security is excellent unless they've just let a load of slow/clueless people over from the normal queues. With a simple set of bag checking requirements (eg no checking across two tickets), you could be airside by 7, certainly 7.15 unless there's a problem.
The staff at checkin will tell you when to go to pre-clearance, and the departure monitors will do so to. There's only one lounge in each terminal, run by the airport itself. As a BA gold flying a oneworld airline you'd get in, fly US in business gets you in too, so no worries on that! Just don't expect a Galleries First style breakfast, think typical European outstation