What sort of construction has been built? Did the neighbour obtain planning permission?
It could be in breach of planning.
The parts which overhang represent trespass. The overhanging parts have no right to be there, then it may be that you can insist on their removal. They are a trespass and, as a last resort, a court has the power to order an injunction to remove them. However, a word of caution is necessary here, a gutter or minor overhanging many feet up in the air may be considered by the court as a trivial trespass. Unless you can show some real harm or serious implication requiring its removal, then the court may not order its removal and can penalise you for bringing a trivial case, penalty being costs awarded against you.
So unless there is a serious problem, you should consider whether to simply let it be.
I do not feel there is more that I can add on the garden room. There is trespass of the overhanging parts but it would be down to the court to decide if the infringement is sufficiently serious to require an order be granted against the neighbour.
Boundary fences are also uncertain. Unless the deeds specifically state who has responsibility for that part of the boundary and us under a specific obligation to erect and maintain then there is not much you can do to force a neighbour to reinstate the fence. If the neighbour refuses then your only option is to reinstate the fencing yourself.