It looks like sprouts coming up from a stump, but a better picture would make ID much easier.

I guessed you were in the UK from your use of the word 'garden' ;-) 1 & 3 are the arborescent (flowering) stage of Hedera helix, common ivy. You do not need to make an application to remove this. It is not a tree. And nor is invasive here. It is a valuable native species but a bit of a pain in a garden. 2 & 4 show a Privet hedge. Again no permission is required as it is not a tree. None of the pictures show Viburnum tinus and even if they did that is not a tree either. All you are doing is cutting back overgrown hedges and you do not need to get permission for that. I also live in a conservation area so I do understand the situation. However, the ivy appears to be partially growing up a tree next door so when you cut it back ensure you are only cutting the ivy and not the tree. If you want to remove branches of the tree get some closer pics and we'll tell you what it is.

Agree w/smivies -- a classic Shumard oak leaf.

It appears to have fruit in the upper right quadrant of the photo and they appear to have wings, perhaps sort of triangular in cross section. So unless I am mistaken, they aren't mulberry. The seeds sort of resemble my Halesia, but the leaves don't look quite right for that. LisaD, can you add a photo of the seeds/pods?