Old English nēahgebūr combined near and dweller. Its spelling changed into neighbor while the basic idea of someone living nearby remained.
Neighbour is the chiefly British spelling of the same word. German relatives are not borrowing stages in this English history.
Sources
- Merriam-Webster: neighbor
Word History and Kids Word Origin explain both components, the continuing meaning and the near/nigh relationship.
- Merriam-Webster: neighbour
The entry explicitly identifies the British spelling variant.
- Online Etymology Dictionary: nigh
Nigh continues Old English *neah*; near was its comparative