From the London area, the easiest (direct flight) options are
Ryanair from Stansted to Pau, or
Easyjet from Gatwick to Biarritz.

Alternatives are .....

Easyjet from Gatwick to Biarritz or Toulouse
Easyjet from Luton to Bordeaux
Ryanair from Stansted to Pau, Lourdes or Biarritz

EasyJet from Stansted to Bilbao
Iberia from Heathrow to Bilbao
RyanAir from Stansted to Zaragoza

(You would need to hire a car from either of these last two airports,
both of which are about three hours away by car.)

(See the AIRPORTS page for specific airport information.)