Dacorum Borough Council states the conservation area rule in its own words: "It's an offence to work on a tree in a conservation area without giving us six weeks' written notice", and explains that "This period allows us to decide if the tree(s) needs protection through a Tree Preservation Order (TPO)". It publishes the exemptions plainly too: "You don't usually need to give notice to carry out work on: dead trees, dead branches and twigs, dangerous trees", with the caveat "However, you must prove the condition of the tree(s). We advise checking with us first".
On orders the council says "A TPO makes it illegal to fell, lop, prune, or damage a tree without our consent", that there are at least 28 days in which to object to an order, and that for the dead, dying or dangerous exemptions "you should notify the council at least five days in advance". Applications go through the Planning Portal or the council's own tree works form. The borough has 25 conservation areas, Hemel Hempstead Old Town among them.
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. gov.uk also states that consent "is needed for carrying out work on diseased and/or dying trees unless some other exemption applies", so a dying or diseased tree is not automatically exempt. Dying on its own does not establish a statutory exemption. We would put the council's page and the national guidance side by side and show which control applies to the work.
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.