I hope you're only chopping down very small trees! Anything larger should be left to professionals.
The council is responsible for taking reasonable steps to ensure the safety of anyone who is working for the council. That includes volunteers (who are typically treated by insurers on the same basis as employed staff). Councillors who do things for the council are volunteers. So appropriate risk assessments should be carried out and recorded, insurers should be notified of any significant or unusual activities and qualified people are needed for jobs that involve machinery or potentially dangerous tools. The work of doing this can be delegated to the clerk, although there obviously needs to be enough time available to carry it out.