You can fly to Limoges or Rodez, or to the newly-opened Aéroport Brive Vallée de la Dordogne with direct flights from London City Airport and Paris (and Ryanair flying from London Stansted from April 2014).
You can drive from any of the Channel ports. Approximate distances:
Eurostar from London Waterloo to Paris Gard du Nord. The Metro to Paris Austerlitz. Then the TGV from Paris to Brive (4 hours) from where you can hire a car.