I dived a bit deeper heheh. So bourne/burne/burn means ‘stream’ in Old English, from Proto-Germanic, this meaning has falled out of usage with this meaning in modern English, and is retained in placenames, i.e Melbourne.
Borne refers to being ‘carried/sustained’, past participle of bear “to bear a great burden”
Bourn in the context of Hamlet means ‘boundary/border’, so only Jesus can pass through the borders of Heaven back to Earth. This comes from French and is thought to be a variant of bodne ‘boundary/limit’
Scots Gaelic bùrn is a cognate (sister, comes from the same root word) with the Old English word and meaning. Like you said, it most likely got to Scots Gaelic through Scandinavia/Vikings and Proto-Germanic.