I would take a slightly different approach that I learned from Mr. Bolton, my guitar teacher eons ago when I was a kid. I've never had a problem switching to a diminished 7th chord on a uke since it's like the guitar that I learned from Mr. Bolton.
THINK left hand chord fingers! 4 2 3 1 (just keep repeating that 4, 2, 3, 1 in your mind or out loud, 4, 2, 3, 1, 4, 2, 3, 1). Then place those fingers on the strings:
4th finger = 1st string, 2nd fret
2nd finger = 2nd string, 1st fret
3rd finger = 3rd string, 2nd fret
1st finger = 4th string, 1st fret
[1212]
There are only 4 diminished 7th chords and they repeat so you can move that same chord up the fretboard and it repeats. Very handy for arranging. Just keep thinking 4, 2, 3, 1 and place your fingers on the strings always starting from the 1st string (4th finger) and the fingers just cascade over each other. It becomes automatic very quickly. Then, you don't need any tricks to jump to [1212] because the diminished chord will be emblazoned into your finger muscle memory forever as 4, 2, 3, 1.
cheerio!
Petey