It's Ragtime, though there's a dependence on how much the piano dominates the arrangement over the horns. Once the horns & woods [clarinet etc] start to take over, then it becomes more Trad Jazz, which some might term Dixieland. [If I remember my musical history correctly, dixieland doesn't have to have a piano at all, it's sometimes banjo instead.]
A 1920's early jazz style, epitomised by Scott Joplin. Its trademark is stride piano where the left hand is busy playing octaves in large leaps & takes quite some skill to perform
Popularised additionally in such movies as The Sting, which also brought the style back to a modern-day audience.
It may not be complete coincidence as the two movies were popular at the same time - JCS was written in about 1969/70 so the producers of The Sting may have heard it used in that & brought it even more to the fore.
Perhaps just zeitgeist.
Just for fun, a bit of live ragtime - this YouTube video is a friend of mine, Paul "Harry" Harris, playing The Beatles' Lady Madonna at Abbey Road Studios... on the original piano.