Shouldn't three of these be inversions?

I not very literate in these things but I thought if the low note was not the root it would be called an inversion. ???

Yes. But they still are all the same chord.

Fully diminished 7th chords are completely composed of minor thirds so any note in the chord could be the root

Wow. So Cdim7, D#dim7, F#dim7 and Adim7 are all the same chord?
Is it the same with E, A#, C# and G and all other variations of the kind?

I downloaded a chord chart for Baritone Ukulele and found that and these other paralells. BTW the strings are the same as the four high strings on a guitar: DGBE. These are also Dim 7th.

I have a junky classical at work and really don't like how wide the neck is. So I use four strings on it. I tend to work on Tenor Guitar sounds or in this case Ukulele tunes.

Try this. Go to the arpeggio page on this site. Choose Cdim7. Don't choose a fret, this will give you the entire fret board. Change the root to Adim7, D#dim7, F#dim7. Notice ALL OF THE NOTES on the ENTIRE fretboard stay the same.