Hertsmere Borough Council states it directly: "In order to work on a tree in a conservation area you are required to give six weeks' written notice describing what you want to do." It sets out its three options on receiving one: make an order if that is justified in the interests of amenity, decide not to make one and let the six week period expire, or decide not to make one and write to tell the applicant the work can go ahead. It states the exemption too: "You do not need to give notice of work to trees that have a stem diameter of less than 7.5cm measured at 1.5m above the ground (or 10cm if thinning to help the growth of other trees)."
On penalties the council says that felling a conservation area tree without consent, or damaging it in a manner likely to destroy it, could mean an unlimited fine, and that other work such as pruning without consent could lead to a fine of up to 2,500 pounds. On orders it says an order makes it an offence to cut down, top, lop, uproot, wilfully damage or wilfully destroy a protected tree without its permission, that a tree that is dead or dangerous cannot be protected by an order, and that where appropriate it can serve an order lasting 6 months before a decision on making it permanent. Its conservation areas page states that there are 15 conservation areas in Hertsmere.
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".
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.