Firstly, try and avoid Heathrow airport, its hard to get you and the bikes there, and the bikes get treated badly.
Ive had good experience at Stanstead, which is easier to get to, and has better staff.
I usually ride my machine to the train station, and then roll the bike carrying all the panniers. This is much easier than carrying the bike yourself.

I try to keep my machine looking like a bike. If it looks like a suitcase, it gets treated like one, and thrown around.
I cover the tubes with plumbing foam insulation, lower the saddle, turn the bars, remove the pedals. I put the chain on the small ring/large cog

At the other end, I reasemble the bike and ride out from the airport. It helps to arrange a morning flight, so you arrive at midday.