Hi Simon, I am UK too. I go by ferry to Santander ... they take bikes ... cabins are expensive but the salons with the overnight 'sleeper' seats (purgatory!) are carpeted so I just sleep on the floor, is only one night.
Spanish trains allow bikes so not too difficult to get it there and back .... but ...
Doing Camino on a bike is accidentally super-fast so you never really get to smell the flowers .. also, at walking pace pilgrims form bonds but on a bike you miss all that as any you meet one evening you will never see again.
So I would say walk .. from Santander or Bilbao (get a ferry that arrives in the morning) you can get a coach to Pamplona or anywhere really - they are comfortable and cheap.
On a budget? Walk for certain! Carry food with you .. sausage, cheese, bread, tomato paste - great nutritious food super cheap whenever you want it at a fraction of the cost of cafes and you can add items daily; tomatoes, yoghurt, etc (carry a spoon and a corkscrew with you!)
I also carry a small amount of olive oil, sea salt, and ground black pepper (I grind my pepper fresh at home before I go as ready ground is horrid) .. and a 500gms packet of pasta. In any refugio with a kitchen I can make a simple nutritious meal.
I also have a pilgrim meal at set price when I want good company or a change.
Mind you, I am not a 'foodie', a food fetishist, I only eat to not be hungry, so am happy to have the same meal day after day.
More budget - if you are a coffee drinker give it up before you go, then you won't need to keep stopping at cafes to get your hit - saves huge amounts over a few weeks.