This breezy farewell always sounded to me like something an American disc jockey would use as a program closer.
I was half right. It was used by a disc jockey, but all the references I've found to it on the Net have an Australian connection:
1. The disc jockey would be Tony Withers, who was well-known on 2SM in Sydney for about ten years from 1953. He was even given a co-writer credit on Johnny O'Keefe's Wild One when it first came out, though his name was usually dropped from later versions.
Tony Withers moved to Britain and worked on offshore pirate stations from 1964, firstly with Radio Atlanta. At Hans Knot's radio site Alan Hamblin recalls that he ended each show saying, "This is Tony Withers, Your Man With the Music, saying 'See you around, like a record' ." Later, with Radio London, he was known as Tony Windsor, and the Pirate Radio Hall of Fame has a Spotlight on Tony Windsor page, with photos and audio clips of his broadcasts.
2. There's also a record called See You Round Like A Record, by Little Nell, a 1976 single on A&M, re-released later as the B-side of Fever. Little Nell is Nell Campbell, an Australian from Sydney, best known for appearing on stage in The Rocky Horror Show and then in The Rocky Horror Picture Show (1975). She wrote See You Round Like A Record with Richard Hartley and Brian Thomson, who produced the track with Phil Chapman.
3. Apart from Rocky Horror sites, a Google search for 'see you round like a record' will tend to throw up links to tabs and lyrics for the Australian song I Was Only Nineteen by Redgum, posted by Armour Byte the Asciizer, whose work is signed Like hip jive dude... and see you round like a record, don't go square like the cover... I kinda like the bit about the record cover: I wonder if that's also from Tony Withers, or did Mr Byte came up with it himself?
4. The discography page of Bob Howe, Australian country music artist and producer, is headed See you 'round, like a record.
5. There is a similar saying, See you 'round, like a rissole. My attention was drawn to this by Air & Angels, an Australian 'bloglet' by La Déesse where a connection is made between the two sayings. Its usage on the Net seems to be largely from Australian sources.
Now I'm wondering whether See you 'round like a rissole is a traditional saying. Which came first: the rissole or the record?