You should understand that whether or not your neighbours agree to the development is not necessarily the point. If Planning Permission is required it is not your neighbours who can grant it. Only the Local Authority can do so. You neighbours may object to it as part of the planning process but even if they all agree, if the development does not meet all the planning regulations then permission will not be granted.
This is why you must engage with your local authority and be guided by what they tell you. If they rule that planning permission is required and you refuse to apply for it (or apply for it and your application is refused) they can and almost certainly will order you to take the shed down. Better to engage with them, co-operate with them and listen to what they have to say. There is no point in arguing the toss with them because (a) they almost certainly know the regulations better than you do and (b) they hold all the cards.
I have to say that I hold no brief for Local Authorities. Their usual response to anything I have asked them for is “No, it can’t be done”. But I will make an exception for my local Planning Department and Building Control. I have always found them very knowledgeable, pleasant and co-operative, often stretching a point or two in my favour when they can. But if you go in with your hackles up the shutters will come down and you will be the ultimate loser.