As it happens in my last job we were in an industrial unit which was owned by a farmer (converted cow shed). When we went digital (got ISDN), BT had to put a new fibre in across the farmer's field and I got to see the wayleave.
IIRC, the term was for 10 years, and after that needed 1 year notice.
If there is nothing in the deeds of the house saying BT have such a right, and you can't get hold of any other documents suggesting a formal wayleave, and you've had no luck getting them to deal with you, then this is what I'd be tempted to do. Clearly this is only to be done if a) you do want it removed, and b) you are bold enough to follow through. I believe BT Openreach as a utility have a right to run services under public highways, but not on/over/under private land. I believe case lore does allow them to run overhead cables that pass over your property (eg to get to a neighbours house)
Write to BT OpenReach giving them a reasonable time (say 12 months) to remove their box from your property - if there is in fact a 12 month break clause in any wayleave, you've just invoked it. Also revoke any permission express or implied for their employees, contractors, and agents to enter onto your land. If they do in fact have a wayleave in place, they should get back to you along the lines of "you can't make us remove it without <some period> notice" - you will now have a person to deal with and hopefully you can resolve the mater with them.
If they just ignore you, then write again and give them another period to remove it, and add that if they don't you will have it removed - and you won't accept any liability for problems caused, and you will bill them for the cost. This is very likely to get their attention.
If you still haven't got their attention, then step 3 is to remove the box. Now this is rather a drastic step to take 9and could make you guilty of criminal damage if you were wrong about the facts), so I wouldn't actually remove the box. I'd go and buy a load of disconnect wedges*, and shove them into their terminal strips. This has the effect of disconnecting the phone lines that run through the terminals - and they'll get a flurry of "my phone has gone off" calls. Trust me, this
WILL get their attention
You need to do this when you are able to stop at home and wait for a man to arrive in a van - probably a number of men in vans. As they head to stomp over your garden, you can then pop out and inform them that they are trespassing and no they cannot get to the box. Explain that you have written to their employer explicitly removing any permission to use your property, and if they continue then they are committing a criminal act**. You can point out that you have attempted to contact their employer, got no response, and have given them notice that you will be removing the box - but so as not to cause permanent damage you have disconnected the lines to get their attention. You can reasonably expect much use of mobile phones and to be put in tough with a manager now who will be very keen to sort it out because a) he's against the clock for repair times on all those out-of-service lines, and b) he'll know just how much it will cost him to have the box moved.
Of course, and alternative would be to just build your wall, embedding their box in it if need be, and if it means their box can't be accessed then ... oh dear, how sad.
* Most "geen boxes" these days are filled with rows of Krone disconnect terminals. These have a slot in the middle into which a plastic wedge can be put to disconnect one side from the other, or can be used for a test plug.
** Trespass is normally a civil tort, but if someone continues after you have told them to leave then the trespass becomes a criminal matter under fairly recent legislation.