G7 under a hand
Strummed from the lowest string to the highest, this grip sounds G, D, F and B, which are the degrees 1 5 ♭7 3 rather than the 1 3 5 ♭7 of the written chord. The root is the lowest note, so this grip is in root position and sounds the way the written chord suggests.
Strummed from the lowest string to the highest, this grip sounds B, F, G and D, which are the degrees 3 ♭7 1 5 rather than the 1 3 5 ♭7 of the written chord. The lowest note is the 3 rather than the root, so this grip is in first inversion, written G7/B. It has the same notes as the root-position grip, but with B in the bass it sounds different.
Strummed from the lowest string to the highest, this grip sounds D, G, B and F, which are the degrees 5 1 3 ♭7 rather than the 1 3 5 ♭7 of the written chord. The lowest note is the 5 rather than the root, so this grip is in second inversion, written G7/D. It has the same notes as the root-position grip, but with D in the bass it sounds different.
Strummed from the lowest string to the highest, this grip sounds F, B, D and G, which are the degrees ♭7 3 5 1 rather than the 1 3 5 ♭7 of the written chord. The lowest note is the ♭7 rather than the root, so this grip is in third inversion, written G7/F. It has the same notes as the root-position grip, but with F in the bass it sounds different.
Four shapes for G7 on ukulele: G7, G7/B, G7/D and G7/F.
Every grip here is G7, but no two of them sound alike. A chord with the root as its lowest note is in root position, and a chord with any other note at the bottom is an inversion. On a keyboard you choose the inversion you want, but on a neck the shape your hand can reach usually decides it for you.
If you add a fourth note, a minor seventh above the root, the chord stops sounding settled. It sounds as though it is going somewhere, and the chord it pulls towards is the one a fourth above it. That is how G7 pulls towards C, and this pull does much of the work of moving a progression forward.
On a ukulele a seventh chord is often the easier grip, not the harder one. You have four strings and four notes, so each string plays a different note and nothing is doubled. G7 needs three fingers and is a comfortable shape to hold.
Because each string plays a different note of the chord, every string matters. If one string is muted or buzzes, part of the harmony goes missing, so once you have the grip, check each string on its own.