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You can remove material on your side of the boundary and the material remains the property of the owner, if the tree is growing on adjacent land. You are not obliged to fling it over, if they don't want it.If the tree is in the neighbour's garden I have a feeling that legally you are required to return anything cut from the tree to them.
We can look forward to some quite vigorous growth over the next few years - and the 'high-level pollarding' makes future cutting much more labour-intensive - lots of little twigs, rather than a few decent branches.I know it won't make the shed better, but the tree will grow back.