I find Google maps to be the most reliable source of information on travel by bus and train.
If you are flying from the UK to Biarritz in spring next year, (i.e. in March, April or May) you will be flying with Ryanair (three times a week, Friday Sunday and Monday from London Stansted) or easyJet (Tuesdays in April,, Tuesdays and Saturdays in May from London Gatwick, no service in March.)
Depending on the time your flight arrives in Biarritz, you have the choice of catching a bus from the airport to Bayonne railway station (bus number 4 from outside the terminal is the quickest and most frequent service and will take you within a few minutes walk of the station.)
Or you can take a taxi, possibly sharing it with with other pelerins (easily identifiable by rucksacks and trekking poles.)
The first train of the day leaves at 06 39, arriving 07 42 and the last one at 18 37 arriving at 19 43. It's a very scenic ride. Check the timetable close beforehand, as there is sometimes a bus substitute. (SNCF website will give you all the details.)
If you won't arrive in time for the last train you have the options of Express Bourricot (no personal experience but widely recommended) or staying the night in Bayonne (several cheap hotels near the railway station, good restaurants and a stunning cathedral) and getting an early start on the walk to Roncesvalles the next morning.
Have a great time!