Q.what type of tree is this?
this seems to have popped up from nowhere, and I have no idea what it is. I live in the UK – London.

Certified GKH Gardening Expert
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