Can the council cut down a tree I plant on common land?
Yes.
When you plant a tree on common land, you are making a donation to the commons and the tree is no longer yours. The fact that it is common land resolves the question. This is really obvious in the fact pattern where you plant a tree in middle of Hyde Park, and is less obvious, but still true, in the fact pattern in the question.
But, be aware that lots of land that is commonly believed to be common land (particularly in the fact pattern in the question), is in fact, private land subject to an easement in favor of the public. There is a fair amount of micro-variation in the common ownership v. private land with easement norm between different neighborhoods in the same municipality, that flows largely from customary practice at the time that particular neighborhoods were developed and deeded (which can span more than a thousand years in some U.K. cities), which has gone back and forth over time.
Where there is private land subject to a public easement, a property owner would retain ownership of the land and the tree, subject only to limitations associated with the easement. But, those limitations might very well authorize the Council to remove or cut down the tree to further the purpose of the easement (e.g. if it was blocking a sidewalk with its growth, or threatened utilities for which the easement was created).
Of course, in some circumstance, the Council can enforce general regulatory laws that authorize cutting down trees on private land (e.g. tree infected with something that could spread to all trees of that type in the area).
Also, it is possible that some public land by be under the control of an entity other than a Council, like a neighborhood association, or another governmental entity (e.g. in the case of a verge along family housing on a military base, or in the case of rental properties on land derived from by a university's land, or in housing associated with a royal palace). In those cases, somebody with responsibility for the land could cut down the tree, but not the local Council.
