I don't know where and when he picked it up, but Ian Stewart was a pianist who specialized in boogie woogie from the 1930s and '40s. Boogie woogie is a term used for the start of the development of rhythm and blues (and eventually rock & roll) out of swing music. Boogie woogie is typically 12-bar blues with a consistent bass pattern that is played with the left hand, while the right hand carries the melody. Pianists like Pete Johnson, Meade Lux Lewis and Albert Ammons had a big influence on Stu, who helped forge the Stones' initial style and sound.
Clarence 'Pinetop' Smith's 'Pinetop's Boogie Woogie' is considered one of the most important boogie woogie pieces to emerge from the late 1920s, and it has influenced every boogie pianist who has followed since.
In his 'Blues Odyssey' book Bill Wyman describes the scene of the time:
When 'Pinetop' moved to Chicago in late 1928, he lived in an apartment house with Meade Lux Lewis and Albert Ammons, both of whom would redefine boogie woogie in the mid-1930s. Lewis did record for Paramount in December 1927, but inexplicably they did not release his recording of 'Honky Tonk Train Blues' until 1929. Recession-hit America was not receptive, and it was almost six years before Lewis got another shot at stardom. He recorded 'Honky Tonk Train Blues' again in 1935; it is actually very similar to 'Pinetop's Boogie Woogie', perhaps because the two men had regularly jammed together.
In the second half of the 1930s Lewis, Albert Ammons, and Pete Johnson became the rage. Ammons first recorded in January 1936 with his Rhythm Kings. Pete Johnson's first sides were made for the Library of Congress, along with Ammons and Lewis, in December 1938. This was five days after the three pianists had played together at John Hammond's ground breaking From Spiritual to Swing concerts in New York City's Carnegie Hall. On stage the three boogie masters performed 'Cavalcade Of Boogie' together, and quickly brought people out of their seats.
It was the beginning of boogie woogie's golden era, which lasted into the early 1940s. For the next three years, all three men recorded regularly, sometimes together. Ammons' signature piece was 'Boogie Woogie Stomp', while Pete's was 'Roll 'Em Pete'.
Source: Bill Wyman, Bill Wyman's Blues Odyssey, Dorling Kindersley Limited, 2001.