Uzes can be easily reached by budget airlines from the UK.

Ryanair fly to Nimes which is only a 30 min drive away and Flybe and Jet2 fly to Avignon which is just a 45 min drive away. For those not wishing to hire a car, Taxis are an alternative method of getting to Uzes from both these airports.
Marseille and Montpellier airports are slightly further away (approx 1.5 hours) and are serviced by a number of airlines.
Athough further away, both Carcassonne and Girona and reached by Ryanair and due to the very good road network, can also be considered.

As an alternative to flying, Nimes and Avignon are also main TGV stations.