this seems to have popped up from nowhere, and I have no idea what it is. I live in the UK - London.
It appears to be Rhamnus Cathartica, or the Common Buckthorn. Removing them before they seed will be important to control them. Then killing the roots will be in order. This article will help you to kill invasive trees and their roots: https://www.gardeningknowhow.com/ornamental/trees/tgen/how-to-kill-a-tree-killing-trees-in-your-garden.htm
I have two fineline Buckthorn plants that are well established. This year, they are majorly affected by rust fungus and there is way too much to remove all infected leaves. Can I cut back the whole plant and will it survive if I do it at this point in the season? (I live in southern Wisconsin--It is a zone 5 here)
Treatment will need to be done with a fungicide. Cutting infected growth will help speed things up, but be sure not to cut more than 1/3 of the shrub at once. Also, it is important that you do this during dormancy, or when you notice that it is not actively growing.
Here is are some articles that will help:
My garden goes to a point at the bottom and I want something to hide the fence of the neighbours garden but I have limited space no wider than 60 cm
Here are some options. One not mentioned is Fine Line fern leaf buckthorn.