Make it easy for them ... offer to build a new fence to demark the area, suggest that you might be interested in extending your garden and cultivating it, don't mention the apiary and start off by asking if they would be willing to transfer the area to you if you paid for all the legal fees and the Land Registry changes (you can expect a bill for that of around £1200) plus the cost of the new fencing ... not cheap. I would not start with an offer to buy it - that's your fall back position when they say 'what are you willing to pay for it ?'
With no opportunity to build on it and no other access it is largely worthless and you could reasonably point this out (I assume they are not using it ?). So .. first offer if they reject a straight transfer - low hundreds I think.
If you don't have the deeds to your property and next doors it might be a good idea to buy them from the Land Registry - you can do it on line - costs pounds. It will tell you the true boundaries, what the mortgages are if there are any and who definitely owns it ... you need to know this because if there is a mortgage on their property the mortgagor will need to give permission for the transfer - another fee for this usually.
Don't go near surveyors or estate agents, don't tell people what you are planning - anyone - the less interest you express the better - make it look like you are doing them a favour when you broach the subject and try not to look too eager.
How do I know all this ? ...I tried to buy the bottom half of my neighbouring garden many years ago ... it was wildy overgrown and the house was rented out and the tenants used it as a dumping ground. It was all going fine until someone suggested to him that land in the area was £30,000 an acre (at the time - flat, building land with no trees and the potential for planning permission it was !) but with no access, a large sycamore, a yew tree and a pine tree all with TPO's on them his 200m2 was not .. he upped his price to more than I wanted to pay and I said no. He moved a few years later and the new neighbour who lived there actually looks after it.