What a mess this whole thing is.
I agree with the general premise of what a couple of posters have said already:
1) The owners of the Station Cottage genuinely believe they own the platform/track area. It is extremely unlikely that they do own the platform, and they have probably misunderstood what they were actually buying, or have made some incorrect assumptions about what they own, based on ancient documents.
2) The land they claim to own which is the other side of the driveway (the area with their washing line and the new Network Rail path) isn't theirs either. They claim to own it through what they say is 'Solomn rights' (a term that doesn't actually exist) but what they mean is some form of Adverse Possession (called Positive Prescription in Scotland) but that isn't possible either it requires them to use it for 10 years then go through a legal process to claim it as their own, which they obviously haven't done as they have only lived there 3 years. The previous owner of Station Cottage obviously hasn't legally claimed the land either. Again, this is likely a complete misunderstanding on their part as to what they legally own.
3) The issue of Network Rail accessing their driveway with vehicles is less clear, as the driveway may be in their ownership. But you can't rely on the satellite maps as they are inherently inaccurate and a qualified land dispute surveyor would need to visit the property in person to make a determination. Additionally, even if they do fully own the driveway, then Section 14 of The Railway Regulation Act 1842 gives Network Rail the power to access and cross private land if they need to carry out repairs and maintenance, or for a safety issue. Also... a court can force access to any land on behalf of Network Rail if the landowner refuses to play ball.
In short, my guess is that they are going to get nowhere with any of this, and will lose badly, then get slapped with tens of thousands of pounds in legal costs if they get taken to court by Network Rail.
I just don't understand it. If it was me, it wouldn't bother me at all to allow Network Rail occasional access through my driveway with vehicles as long as I had a working platform at the end of my garden. On balance, the rail service would be far more use to me than anything else.