Everything I have read so far does not indicate a training time other than materials I found in the Emerald Empire sourcebook with the chapter discussing education. It vaguely says that getting the necessary skills takes 6mo - 1yr depending mostly on the student.
Other than that, even sections about skill training in the Book of [Element] sources are more blurbs about the methods and goals than anything to do with time (as I imagine they are more trying to justify why the schools are different). Even a section on training war horses in Sword and Fan followed this pattern.
The only speed of growth I could find were sections on gaining/spending XP. I looked in the core books for the editions 1-4, and the GM Survival guide (AEG 3001) and their only points that can be assumed to be about time are the rate the GM provides XP which can be (by default at least) spent immediately if desired. The closest to an answer I found was in the 1e Core, pg 61 under "School Rank"
When a Samurai or Shugenja wish to gain a new School rank, the character compares his current Insight points with how many he needs to gain a new School rank. If the character has enough Insight points, then the character retursn to his school, and the sensei must be show his progress. If he is impressed, he will teach him the secrets he needs to perform at the new Rank. In addition, the character will gain any special abilities conferred for increasing in rank. Some Schools will also insist that the student pass a test in order to prove his worth. This test could include something as simple as a written exam, while other will demand more esoteric tests of the character's skill".
The 2e core says on page 87
Increasing a character's Trait or Void above 5 requires a significant amount of roleplay outside of the experience expenditure. Typically when a character spends the experience to raise a Trait or Void, they must undergo a series of 'tests' (given by their sensei, some celestial power or other appropriate GM NPC) in order to actually achieve the new level.
The GM Survival Guide says on page 55
A Strength 2 character that becomes a Strength 4 character after two months of play is exciting for players and, although unrealistic, has merit in a roleplaying game, where the GM and players are focused on 'fun'
As a personal aside, I have been a fencing coach for a number of years and I can honestly say there's no perfect formula. Yes, you can have classes and boot camps and things to get a minimum understanding of the skills and some fitness which can account for the first few ranks of the skills, but it's a bit hit and miss to determine whether someone's ability is linked to skill growth or a naturally high trait floating their 'rolls'. This is why I like L5R's method of "leveling up" (short of 5e which has a WHFRPG style checklist) with insight because the breakthroughs we make can be tied to so many other aspects while we grow as people. It's more than the sum of our parts. I would make metaphors to things like basketball, martial arts, music, acting, etc. because they were concepts that the student could link to their understanding of fencing.
As a coach I would require several hours of footwork drills before I even let a student hold a weapon, no matter how prodigious they were because there are things like muscle memory that need to be developed over time and practice. Some people, while earnest in their journey, only practiced within the club despite a number of workouts and drills they could do individually or that they could meet with fellow students and spar between club meetings. Even still it would be an individual journey. Those first few ranks are easy to obtain; the rest require practice, study, time, and dumb luck of Experience.