One of the critical little make-or-break things for any instrument is fingering.  If you can't push all the needed keys/buttons/valves fast, consistently and easily, the shiny new instrument "jest ain't gonna make it".
In early days of jammer research, the worry was small, but nagging: in any design of something new, there's always the chance something was overlooked.

We're happy (actually we are totally relieved) to report no show-stoppers: the fingering is easier than the piano's, and perhaps the easiest of any instrument yet developed.
I've also found that some accordions and concertinas use this fingering, Further, all the world's fastest accordions players use this related C-System, so it the theory does indeed translate into practice.

But! ... Its different.

This what I recommend, shown at the right.

This fingering works for both relative minor and major scales. It's simple and almost always works - the fingers seldom conflict with each other.
The index finger and the little finger are also easily able to reach notes that come up frequently in nearly all the popular music pieces tried, the Bb (Minor 7th), F# (tritone), and even C#. This agrees with what music theory predicts.

The other accidentals take a bit more care, but reaching for them is becoming automatic and easy.

I don't recommend ...
Putting the index finger on the root seems more natural at first. However I found this causes many more fingering problems and it's hard to reach the F and Bb.

A seminal moment
Recently my wife (an expert pianist) was surprised that I could play a rangy, two octave song on the first try. On a piano, a novice of my skill level would have to watch one's fingering most carefully. On the jammer it was a no-brainer.

Downsides?
Naturally, there are some. 

• Occasionally, the fingers will end up so that there is not a handy finger free to press the next note without having to use a finger already holding a note. This causes a slight gap in the playing, but this happens much more often on the standard keyboard.
• Sometimes a whole group of notes outside of the key center pops up. In this case, it's usually easiest to move the whole hand, and pretend the key has changed. With the limited number of keys on adapted Axis-49s, this can be a challenge. 

More: full keyboard fingering