I've been thinking about your post awhile, since BBC America first ran this show. They will be showing it again on Tues., Dec. 26, 12 midnight-3 a.m. (Pacific time), so I finally decided to reply. One song played in the show is a standard called "I'm Getting Sentimental over You." As I recall, this is played while Bob is walking through a park. Another song I think was played is "Red Sails in the Sunset." The melody of the song you asked about--the one that is played repeatedly--was familiar to me, but I couldn't put a name to it until I heard it recently on satellite radio. It is called "Goodbye" and was performed by the Benny Goodman Orchestra. There was no vocal, and I don't know if there are words to it.
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