Did he do exercises, or just work on repertoire and improvisation? Are there any anecdotes that may shed light on Tatum's routine?
Everything I think I know of Art came from my teacher (whom I won't insult by naming her) who studied with Oscar who of course knew Art. So what I can remember, fourth hand, she opined that Art was one of those rare people who had a convergence of several factors whereas any one of us may have one or two but Art had them all: technical facility, harmonic genius, melodic, rhythmic, a sense of arrangement with an underlying desire to catch the listener off guard.
First, he had incredibly large hands yet slender fingers so his size didn't get in the way with extraneous adipose tissue. He could reach notes that most people couldn't and never will. Secondly, he had perfect pitch and just knew what the notes were of anything he heard whether aurally or in his own head. He would listen to his sister practice something and would come downstairs and just play it flawlessly from hearing her. This was confirmed to me when I was ballsy enough to call Oscar's sister Daisy for an autographed picture of her brother. She had the best answering machine messages.
In the beginning, Art proclaimed to be self taught and happened to be one of those rare (prodigious) people who moved properly and ergonomically right from the start and didn't have improper movement or bad habits to get in the way like many of us normies do. Because of this, he didn't have to work on technique. Much like you don't have to practice walking, it just comes naturally. You probably don't realize that in order to walk forward one foot is pushing backward and your shoulders and hips are working in contrary motion and rotation which protects your knees. The same thing goes on in our arms when we play, or, for some of us, it doesn't. The difference between Art and us is that he totally moves properly and we struggle to figure it out and battle bad habits ingrained in our brain's muscle memory from our first formative lessons. The first teacher had better be the right one.
Apparently a bulk of Art's practice was working out arrangements from which within there would be "improvisation." I put that in quotes because he knew everything he was doing. He was rarely just making stuff up. All his solos were well crafted.
Blindness played a factor. Art loved sports and had he been able to see he probably would have been a baseball or football player. BTW, violin was his first instrument. He had a little bit of vision in his right eye but he could only see straight down which is why at the piano he held his head up so he could see the keys. While walking down the street he would tilt his head way back so he could see forward. But, after a mugging he lost his good eye. Although he had an active social life, there wasn't much he could do other than work at the piano.
EVERYBODY talks about how he played with flat fingers. Both Lee and I disagree that he played with flat fingers. It only appeared as if they were flat. Since he was always throwing in octaves and thick chords, when you spread out your fingers, of course they look flat yet you know your fingers are still curved especially at the distal phalanx. Also since he played from the arm and not the fingers, the fingers he wasn't using were relaxed and went in the same direction as the finger he was using. The finger that he played down was always playing straight down and had a slight flexation to it but people didn't see that because the "flat" fingers were in the way. Also he had a forward shift to most of his fingers which came from the elbow. People who attempt to play with flat fingers often break the fulcrum of the distal or intermediate phalanx or develop long flexor tendinitis so don't try it at home unless you play from the arm. That is what gave him his speed because he never isolated his fingers and his playing actually came from rotating or tumbling the fingers in groupings. Oscar had a word for it but I don't remember what it was, maybe slapping?
I don't know if any of this helps. But to answer your question, I don't think he had a practice routine, he just played. Again, much like you don't have to practice walking because it comes effortless to you, he didn't have to practice technique because it was natural and effortless to him. It just wasn't something he had to work on. Lee said that he never practiced but I'm sure prior to his mid twenties he spent a lot of time figuring stuff out. When he was invited to NYC he didn't think he was ready for the big time.
Oh, another thing Art had going for him was that he was a night owl. He loved playing in the local speakeasies and after they closed at two or three in the morning he would go play at the after hour clubs until five or six in the morning. Everyone knows that one performance is worth ten rehearsals and Art performed seven nights a week for a good nine or ten hours.
Fact or fiction? I don't know. It is too bad there isn't more footage of him playing but Art wasn't a musician of the people, he was a musician's musician and sadly wasn't as famous or as well paid as lesser pianists. I do know this for a fact that one of Art's regular sidemen said in an interview that Art rarely practiced.
A lot of people attempt and some come close to playing like Art but they never totally succeed. Art was a one in a million fluke amalgamation of all the right stuff. There will never be another Art and that is a good thing. There is a pianist who spent his life trying to emulate Oscar. I forgot his name (Ella's pianist) and although he is a million times better than I, I have difficulty listening to him because he is good but not quite there. Even Oscar after his stroke still sounded like Oscar and not a cheap imitation. Like honey and saccharine.