The modern popular use of buns in the sense of "male buttocks" is from 1960s, perhaps from a perceived similarity; but bun also meant "tail of a hare" (1530s) in Scottish and northern England dialect and was transferred to human beings (and conveniently rhymed with nun in ribald ballads). This may be an entirely different word; OED points to Gaelic bun "stump, root."
"scrawl aimlessly," 1935, perhaps from dialectal doodle, dudle "fritter away time, trifle," or associated with dawdle (which might be the source of the dialect word). It also was a noun meaning "simple fellow" from 1620s.
LONGFELLOW: That's a name we made up back home for people who make foolish designs on paper when they're thinking. It's called doodling. Almost everybody's a doodler. Did you ever see a scratch pad in a telephone booth? People draw the most idiotic pictures when they're thinking. Dr. Von Holler, here, could probably think up a long name for it, because he doodles all the time. ["Mr. Deeds Goes to Town," screenplay by Robert Riskin, 1936; based on "Opera Hat," serialized in American Magazine beginning May 1935, by Clarence Aldington Kelland]
Related: Doodled; Doodling.
Doodle Sack. A bagpipe. Dutch. — Also the private parts of a woman. ["Classical Dictionary of the Vulgar Tongue," 1796]