Well I do know how these things work in England and, no, you wouldn't be able to do this in England & Wales without the consent of all the others.
No idea how this works in Scotland, but it seems unlikely to me. The point is you could drive over their sections of owned road, even though it is of little use to you.
However whatever happens, the cheapest way for you - and you could argue for everyone (if you can't do your proposed scheme) would surely be to organise a joint repair and maintenance job that everyone pays according to the square metreage of road that each individual owns.
Are we talking big money here? - it doesn't sound too much pain.