I misspoke. You're right. I meant I need to remember what chords my old chord shapes have become. When I play on my own, it doesn't matter so much, but when I accompany the kids on violin I have to pay attention to that.
Coming from guitar, when I first started on the uke, it was a real struggle for me to unlearn the chord names and just focus on the shapes, and then the names came later.
I must confess that when I play on baritone or other ukes not tuned GCEA but with the same modified-fourths tunings, which I do mainly to change the key for my baritone vocal range, I dont transpose at all, if the music calls for a C chord and I am tuned down 2 semitones to the Bb tuning (F-Bb-D-G) I dont bother to notice that the SHAPE of the C chord is actually a Bb, but if you 'ask' me, I can tell you it is later on...
I do the same thing with music written for GCEA with the baritone tuned in DGBE, it is just a 5th lower, and many other folks here on UU have also confessed to doing the same. If you are playing solo most of the time, it's not an issue, but of course like you said, if you play with others, you need to transpose in your head and realize that the GCEA 'C-chord shape' is now a 'G-chord shape' if you are tuned to DGBE in G6 tuning, etc...
Now without much hesitation, and based almost completely on my 30+ yrs of guitar I can 'think in G6' when I play baritone since it's the same as an E-A-D-G-B-E tuned guitar without strings 5 and 6, so really a G chord on baritone is the same as a G chord on guitar, just without the lower bass strings...
Eventually it all becomes kind of automatic. I focus more on the experience of the sound and physical technique rather than naming all the notes on the fretboard, or knowing all of the modal scales in all 12 keys. I rarely use it, and when songwriting, if a minor scale is not what my muse wants, sometimes it is 'melodic minor' or 'harmonic minor' and those I can find by ear to go with the chords...
But do not worry if this all sounds complicated, it will come in time, and the uke and playing music should be a source of joy and not feel like hard work. Music Theory can come later if you prefer, and it's really all up to you to find your best path.