A report and some video clips of Thompson's appearance in the "Great Guitars" series can be found at
http://www.grammy.com/news/richard-thompson-at-the-grammy-museum
Check that footage of him doing the Chet Atkins licks.
I know you're about to ask so I'll tell you.
The Strat is a bitsocaster put together by his roady. See
http://www.richardthompson-music.com/gearandtunings.asp for details.
The acoustic is a Lowden. He has several of these, all different. This one is an "F" body shape and it looks like a cedar top.
This kind of event fascinates me almost as much as a live concert performance. Every now and then I wish I lived in LA.