You can play perfectly well in the key of G using C tuning (or D tuning, or Bb tuning, for that matter). The C tuning doesn't restrict you to the key of C; in fact, I play in any key I choose; I just can't use open strings as often in most keys. But all the open string notes are part of the G major scale, so C tuning fits the key of G fairly well. You just have to play up the neck in order to play an entire scale tonic to tonic.
Retuning A to G gives you "open C" tuning, since the open strings play the notes in the C major chord (CEG) and no others. In open C tuning, you can play major chords with just a barre straight across the frets, making this a popular tuning for slide uke. To play in G, your primary chords would be G (I), C (IV) and D (V), with F (bVII) added for a Mixolydian cast, often encountered in blues, rock and country songs. You can play these chords with barres respectively at the 7th, 0th (nut), 2nd and 5th frets. This tuning is a fourth higher than the most standard banjo tuning (g)DBGD, aka. open G, so the basic chord shapes are the same, although the chord names are different. Clawhammer ukists often use open C tuning to play in C or G.
There are several ways you could retune specifically to G. The "standard" G tuning (used widely on baritones) is DGBE (dogbone), normally nowadays using a low D (linear tuning)—of course, you'd need a different string set to switch from GCEA to DGBE. The major advantages for playing in G tuning is that G is the lowest note, you can play the entire G scale in first position, and you can play each note of the G chord on an open string, and open E is also in the scale of G. The lowest strings, G and D, are good for droning and positioned for easy play with the thumb, as in clawhammer style; they're also the tonic and dominant notes commonly used for drop-thumbing in slack-key styles.
The chord of G contains the notes GBD, so an "open G" tuning that would suit a C-tuning string set is GBDG. The G6 chord contains the notes GBDE, which could be achieved by tuning a standard set GDEB, raising the 2nd and 4th strings a whole step—I personally haven't heard this tuning used on ukes, but it's a distinct possibility, with many of the advantages of C6 tuning in terms of chord construction. G7 tuning would be GDFB, G Wahine (GM7) tuning would be GDF#B. How about Gm7 tuning: GDFBb?
Another tuning amenable to the key of G is an all-fourths tuning: GCFBb, raising the 1st and 2nd strings a half-step. Bass guitar also uses an all-fouths tuning (EADG); you may recognize these notes as corresponding to the lowest four notes of standard guitar tuning. So there's a lot of precedent for using such a tuning, even if it's not common in the uke world.
Other banjo tunings also adapt well to uke. The uke G analogue for banjoists is the key of D. Translating some banjo C and D tunings to uke G, we get:
Open G: G B D G (described above; adapted from banjo Open D)
Open G: G D G B (adapted from banjo Open C)
Double G: G D G A (banjo Double C is also called C Modal)
Standard G: G D F# A (adapted from banjo Standard C)
G Minor: G Bb D G
G Minor Modal: G C D G or F C D G
All-4ths G Minor Modal: E A D G
D Minor Mixolydian: G D G C
WARNING: some of these may be uke string-breakers. Instead you may want to tune a whole step lower, then capo at the 2nd fret. On banjo, to play in D one often tunes to a C tuning then capoes at the 2nd fret (including capoing the drone string at the 7th fret).
