The word can be found in northern England in placenames such as: Redbourne and Legbourne but is commonly in use in southern England (particularly Dorset) as a name for a small river, particularly in compound names such as winterbourne.
The apparent variant, borne found in the placename: Camborne, arises from the Cornish language and is in fact a false friend: it refers to a hill (Cornish: bronn, from Common Brythonic *brunda; compare Welsh bryn).
Born/borne in German also means fount, or spring, and is related to the Indo-European root, *bhreu.
That born/borne appears throughout Europe as a placename is also an important clue that this spelling is an etymological precursor to the Middle English bourne/burn.
Burn (landform), in common use in Scotland and North East England especially, but also found (in placenames) elsewhere in England such as: Blackburn, Gisburn, Woburn, Kilburn, Winkburn, and so forth.