I understand that you wanted to take the pragmatic approach to get the sale over the line on a house you love and with the time pressures of completion and having to move from your current home, @MummyToBeAgain1 . I also understand that you didn't want to risk losing any money, be involved in a protracted process in dispute with the auctioneers, agents and seller, and have therefore negotiated a discount instead. I can see that you want to proceed with positivity and I wish you every happiness in your new home. At the same time:
I need further advice before proceeding.
Yes, you most certainly do.
I want you to ask yourself these very simple questions: Why have the neighbours stolen that particular strip of land? Why do they want and need it so much that they are prepared to blatantly steal it and put up a very substantial and expensive fence to secure it for themselves? It is quite an extreme thing to do and something most of us wouldn't do.
In the cases I have known where similar has happened, it is because they want to build or extend on their plot but haven't quite got enough room for what they want to do. That strip of land juts out into your garden and towards your house, if your diagram is correct. How would you feel if they use it in future to enable them to, for example, build an extension, a garage with a room over, an outdoor bar and hot tub area, a home office or gym, an annexe for a family member or to let out, a workshop, a septic tank, oil tank or heat pump? You might think that you wouldn't mind but the reality is most of us would mind when all those things can result in increased noise, windows opening out into your garden, and loss of light and privacy. What if it turns out that you need that land for something unforeseen in the future?
My advice would be to move in, let the dust settle and get over this unexpectedly stressful buying process. Later on you will be able to seek advice and think more clearly beyond the relief at getting over the line, having your new home and at a discount. Personally I would not be assigning the land to the neighbours via the Land Registry process. As a first stage, I suggest writing to them when you have settled in informing them that you are the new owners of the house, they were observed on x date by witnesses and instructing them to remove the fence and reinstate the boundary within 14 days.
One thing that struck me is that absolutely nobody seems to have spoken to or written to the neighbours about what they have done. They must be laughing up their sleeves.