The straight answer to your question as to whether you could fence it is that you can, but whether you can get away with it from the other neighbours is another issue. Based on what you say, you would fence to fence and maintain it for many years to the exclusion of these to stand any chance of claiming adverse possession.
The planning consent issue is a red herring. There is no connection between applying and gaining consent for a building, and whether the structure can actually be built as it is on another's land. Nowadays a planning applicant has to declare whether or not the land is owned, but it never used to be. And even today it doesn't impact the decision - just the consultation process.
It is perfectly possible that no-one owns the land comprising this yard.
Since your two parcels of land do not adjoining to one another, you should look on your land title document to see what it says about access from one to the other.
Finally, how long have you lived there? The function of your solicitor on buying is to check you have legal access to what you are buying, so even if it is a long time ago, you could complain through that process if it seems this was fouled-up.