UK Property Law
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They have said that there is nothing that they can do. So you cannot beat a resolution from them with a stick. Clearly moving the fence is impossible because someone bought next door.
The remedy appears to be moving the conservatory. Is that what you want?
It seems to be the only solution.
If they simply refuse you are faced with getting a quote to move it sideways and sueing them for that cost.
When legal proceedings arrive on the doormat they may be more co-operative. I wouldnt be hanging around becuase it will be easier to get this done while they are on site. It isnta DiY job and I would see a solicitor and get get a letter off ASAP.
Can I help further?