The most convenient way to travel to Ardèche is by flying to the preferred airport of Lyon-Saint Exupéry. From the UK, direct flights depart from London Heathrow while flights that require one stopover depart from a selection of airports including Birmingham, Manchester, Glasgow and Edinburgh. From Ireland, typically there are no direct flights but departing flights from Dublin and Cork fly to Lyon via Amsterdam, London or Paris.
If you have the time and a reliable car, you can drive to Ardèche after crossing the English Channel via the usual ferry crossing routes from the UK and Ireland to France.