A virus which attacks Willows is unlikely to attack Viburnum as well. Fungal attack could get both though.
Picture of the Viburnum would help perhaps.

You re legally entitled to cut off any thing from yhour neighbour's garden that overhangs yours. Just a thought, though it doesn't make for good relationships with your neighbour...

I just removed a large tree from my own property. I'll tell you that it's intensely laborious, dangerous, expensive and messy under the best of circumstances. It's not a job for the weak, frail or sickly. He/she will have to call in the pros to remove the tree - expensive! I know, but sometimes it has to be done. Give your neighbor a courteous reminder of the potential hazard of not tending to the dead tree. It's their responsibility. When the tree is finally gone, aww! what wonderful relief!

Can you save your viburnum? Depends how far gone it is but that's another conversation.