OK, I am finally getting the "traditional" E chord shape. I realized that my fingernails were getting in the way of me getting a good vertical approach to the strings, so I've started cutting my left nails back to nothing. That has allowed me to get my fingertips almost perpendicular to the fretboard, which really helps with cramming them all into that one fret. I've also found I need to rotate my wrist clockwise a bit from my normal position to help get my fingers "lined up".
I run out of room to do it around the 5th fret on my concert uke.
In general, I'm using a better hand position for all my chording now, and trying to defeat this E helped. I'm keeping my thumb further down on the back of the neck, which makes my hand stand off from the fretboard better. I'm getting far fewer cases of notes being dulled or deadened by fretting fingers "touching" adjacent strings. Practice practice practice!