Code powers can be applied to private land, but it's far from straightforwards. A utility with code powers can make a request to a private land owner for access and wayleaves. If this is not agreed to within 28 days, then the utility can go to court where all the issues are considered.
Of course this whole exercise is very expensive. BT does, of course, have a USO for providing phone lines, and this can involve crossing private land. However, the costs have to be deemed reasonable (which is usually taken as around the equivalent of 2 week's manpower for any one line, although I don't think it's hard and fast). Beyond that, the customer will have to pay the excess. Of course if it comes to legal action to get court approval for access, then that cost could be huge. Not just that, but if the estate isn't pre-built with ducting it could be very expensive.
So there's a potential legal route to get an OR line down a private road, but it's not going to be easy or cheap, especially if there is opposition.