East Herts Council states it in its own words: "You must give six weeks' notice before carrying out work to trees in a conservation area that are not protected by a TPO. This is achieved through a Section 211 notice." It sets the threshold on the same page: "A section 211 notice is required when performing work on trees with a trunk diameter greater than 75mm which are in a conservation area which are not subject to a Tree Preservation Order (TPO)." It then makes the point that catches people out: "The council can reply by making a TPO or by allowing the work to go ahead. As it is not an application for consent, it cannot be refused or accepted with conditions."
On exemptions the council says "Consent is not required when a tree is dangerous or dead, however, you should be prepared to prove that the works fall within an exemption." It also states that "Tree work is a dangerous activity and should be carried out by trained, competent and appropriately insured arborists", and that it can prosecute where work is done without consent on a preserved tree, without the required notice in a conservation area, or where exemptions are misused. The council states that there are currently 42 conservation areas in East Herts, and publishes a Hertford Conservation Area Appraisal.
Nationally, gov.uk describes the requirement for "people to notify the local planning authority, using a section 211 notice, 6 weeks before carrying out certain work on such trees, unless an exception applies", and says the work may go ahead sooner where the authority gives consent. The same guidance states that consent "is needed for carrying out work on diseased and/or dying trees unless some other exemption applies".
On felling, the Forestry Commission service states that you need a licence to fell growing trees in England unless an exemption applies, that the threshold is an over bark diameter of 8cm or more measured at 1.3m above ground level, or 15cm or more for coppice or underwood, and that subject to the applicable exemptions a licence is required above five cubic metres per calendar quarter, and that the five cubic metre exemption allows no more than two cubic metres to be sold. Licences are free in England. Trees in private gardens are among the listed exceptions.
On wildlife, gov.uk states that you are breaking the law if you "intentionally take, damage or destroy a wild bird's nest while it's being used or built", and that "All bat species, their breeding sites and resting places are fully protected by law", damaging or destroying a breeding or resting place being among the offences. We would put that on the page with both links beside it.
We would explain how you check the conservation area boundary, any tree preservation order and the proposed work before confirming whether a section 211 notice or consent is needed.