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Good evening, welcome to just answer. I am a solicitor and I am reviewing your question for you.
Hi, thank you for your patience, Yes you need authority to do anything with trees in another's garden especially if cutting. You do have the whatsapp message and he did not qualify his message so you can treat it an unfettered permission and hopefully you did not "kill" the tree. He will be able to assess when he returns and you can take a view at that stage. Any clarification please do not hesitate to send your follow up message. All the best.
You are welcome, You would not be completely in the wrong where you used a professional tree surgeon and their professional advice was to go as far as they did. I think it is best to not worry in anticipation and see what he says when he sees it in person.