They are all lovely - but quite different. Avignon is definitely more Provencal in 'feel' - and there are a couple of sites there, the one David mentions and L'Ile des Papes (which takes both Camping Cheques
www.campingcheque.co.uk, or the ACSI card
www.campingcard.co.uk from 25th August. Don't worry about 'full on' sites - at the end of August some sites become very quiet indeed. From either you can explore quite a wide area of Provence and even the Camargue - we stayed south of Arles and have gone up into the hills north of Avignon several times from there. Arles itself is a nice town, but is in Camargue country, which doesn't suit everyone though Les Beaux, Les Alpilles, and St Remy de Provence, plus the roman ruins at Glanum are all definitely worth a visit. We also went to the 'Courses Camarguaises' - bull-fighting where the bull doesn't get injured, but has 'ribbons' removed from his horns by the local teams - it was free and definitely worth seeing (though I'd never go to one of the 'Ferias' which are the awful ones where the bull is killed).
Nimes is different again - you're moving up into the Garrigue. Its a lovely town, with lots of roman buildings, a nice market, and a very cosmopolitan atmosphere. From there you can access the hinterland of the Ardeche, etc and the Pont du Gard. There is a municipal site in Nimes, (not recommended) and several sites to choose from near the Pont du Gard but you might like to take a look at Camping Mas de Reilhe at Sommieres, which is a lovely Camping Cheques sites, (not a large site) which takes Camping Cheques from 16th August (
www.campingcheque.co.uk) and does 7 nights for 6, and offers fully serviced pitches for no extra charge after the 26th August.
If it were me, and I couldn't choose, then I'd be inclined to do a few days in each area - which is very much easier on French sites, especially at the end of August, you can just tour as you please, with no advance booking - and if the site, the weather, or the area doesn't suit, you can simply move on after as many or as few days as you wish.