Garden Shrub Removal Rights

Summary

If a shrubs roots and trunk are on your property in england, you can legally remove it. Neighbors can trim overhanging parts. Discussing with neighbors is advisable for good relations.

Full Conversation

No. It may be good practice to discuss things with reasonable neighbours, but less wise to invite the opnions of unreasonable ones!
It sounds like you've got a handle on the legal aspects of this situation. If you need more details or have another legal question, feel free to ask. Maintaining good neighborly relations can be tricky, but legally, you're within your rights regarding the shrub.