10 day walk from SJPP

Best airports from London for easy access to StJdP would be Bordeaux or Biarritz.

10 days including connecting travel or 10 days walking gets you to either Logrono or Santo Domingo de la Calzada both with reasonable transport hubs to get you to Bilbao and a flight back to London.

www.gronze.com & www.rome2rio.com are useful planning sites.

There's a couple of ways you can go:
London - Madrid and then take the train and/or bus to Pamplona and then a bus or taxi on to SJPDP. Easy to get cheaper round-trip flights to a major hub and easy to get public transport to Madrid on your way home.
London - Biarritz and then a local train to SJPDP saves you time arriving into SJPDP however will be slightly more inconvenient getting back to Biarritz to fly back to London however not so bad if you're only 50-60kms outside of Longrono when you wrap it up this year.

As usual, it comes down to the wrestling match between time and money. More money often translates to quicker trip to your starting point or you can take the milk route and save some money but often sacrificing time, which is precious is you're working with 10 days total.

Good luck with your planning!

I've found Bilbao (flights from Heathrow with BA) a good place to get to anywhere on the Burgos side of the Camino. Alsa buses are only two hours to Burgos, Logrono, Pamplona, then a 30 minute shuttle to the airport. Trains are pretty good too.