Inversions can very much be part of ukulele music, but there are some things to remember:
1. If any other instrument is playing, that will almost certainly determine the inversion, as the inversion is simple telling what the bottom note is. So, any note below middle C will determine the inversion. So, if you want the ukulele to determine the inversion, you probably need to play solo. Or with flutes and oboes...
2. The power of inversions lies in the effect of the bass note on the rest of the chord, and how harmonics above that note create tension. Because of acoustics, different inversions have different characters, especially 2nd inversion chords (with the 5th of the chord in the bass). Because ukuleles are such high pitched instruments, the effect of inversions is greatly reduced, as a lot of the tension created happens above the range of human hearing.
I find it much more useful in ukulele composing to think about voicing; that is, which note goes where, and how close different notes are to each other. Ukuleles tend toward very tight voicing given their tuning and limited range. Again, their high pitch suits this really well.