It would probably kill a tree of this age to be so severely pruned, even at the right time of year. (July-October) Would it be better to offer to help re-site his greenhouse?
Explain to your neighbour that is is not reccomended to prune established walnut trees, he may understand. If not then I would get experts to do it.(he may share the cost)
As to the ivy, it may well have birds nesting in it at this time of year. To wilfully destroy a nest with eggs or young in can lead to a fine of £2,000 and/or a term of imprisonment. Good luck. It's only in the last twenty years I have lived close to other folk and I still can't get used to it!