Boo,
Something has occurred to me. I was in Bamburgh churchyard recently doing research of my own. If they lived in Budle it is entirely possible they were buried in Bamburgh church. My husband had relatives still being buried there in the 1980's. The graves are a motley bunch but quite easy to navigate because they are more or less sectioned according to age, the older graves from the 1700's at the front of the church yard and the newer ones towards the back of the church grounds. My husband is pretty sure that if your relative died in Budle that it would be Bamburgh churchyard they'd be buried in, unless they had a family connection elsewhere.
On most documentation I've seen they list Budle as part of Bamburgh, it's literally a couple of minutes up the road and Bamburgh people tend to be buried in Bamburgh and Seahouses people buried in Seahouses. Both villages see themselves as completely seperate cultural identities and even now, villagers of both are proud if they can say several generations came from their particular village. My FIL has always been proud of saying they are "Bamburgh folk". Another area maybe to look is Belford as well, but I don't know when they stopped burying people in St Marys Church and moved to the civil cemetary.
And seriously, if you find out what churchyard I really don't mind - we live within 5 miles of Budle as it happens. The apples never fell far from this family tree!