This discussion has been the maddest I have seen on Mumsnet thus far. WTAF.
Human lives trump dog lives. No exceptions. To suggest that a pet means as much to someone as their (only) child to its parents is completely out of order. Despite what we say, 'fur babies' are not the same as human babies. A dog cannot be equated with a child, no matter how much its owner (not its parent!) loves it. You can purchase (even directly select) a dog at any stage of your life. (The Queen alone has had dozens and dozens of them.) You cannot go out and just 'get' a new child.
I grew up with a dog. But if I had to chose between the dog and the human family members, I would chose the humans without a second thought. In fact, almost everyone in their right mind would. In WWII, my grandfather's family, who were occupied by the Nazis at the time, had no food. With a heavy heart they slaughtered the family's pets. They didn't consider cannibalising a child.
If my neighbour had told me their child was severely allergic, I would not have bought a dog. Even if I had always wanted one. It is clear from the OP's comments that this is a serious allergy, and not 'played up'. Her parents (and entire family) have consciously created a 'safe space' for her to grow up and play in, and this safe space has now come under threat, because of OP's lifelong 'dog dream'. Of course the parents are anxious, the new neighbours have just massive amped up the danger level in the one area of the child's life where they thought they had completely safeguarded her.
Plenty easy for people to say 'they should move to detached/more land'. OP already wrote that the gardens are decently sized. Not every family has the money for a bigger place. In addition, for a girl who sounds very isolated already from others, are you really suggesting to cut off even more ties? Her family live around her dog-less for a purpose.
Also kind of cheap to say 'she should stay indoors'. We would call that child abuse in any other circumstance, but interesting to know that seems to be acceptable to so many PPs when it comes to inconveniencing a dog (who really doesn't care where it plays and does its business).
For those saying disbelievingly 'she wouldn't be able to go anywhere?': that is the whole point. This child does not have an alternative to her garden. It is clear from OP's writing that her world is very, very small, and her life very restricted. Just because you haven't ever come across someone with such a severe allergy, doesn't mean it can't exist.
In terms of OP's suggestion that you never know what you are going to get with a child, it is clear from PP's comments that having a child with an insanely rare and severe allergy is not a thought that comes up in most people. I don't think this is an argument at all, OP. This is clearly an extreme situation. If we had to take all those eventualities into account when deciding on pregnancy, nobody would procreate anymore.
Of course they cannot legally 'force' the OP to not buy a dog and leave it in her own garden. But I think, morally, what OP is doing is wrong. (And I am sorry, OP, you sound nice, but I seriously think you are morally in the wrong here, even if you are legally in the right.) Imagine being in their shoes. I would have been deeply frustrated, too, especially as you got along previously. Why would a 'friend' consciously endanger your child? Even if it is the smallest risk, why would you add another stone to their already heavy burden?
I think that this debate has clearly highlighted that according to the average Mumsnet user, Britain's two sacred pillars, personal property and canine lives, should be protected at all cost. Even if it is an innocent little girl's health and wellbeing that is at stake.