This issues is very fully covered in the thread linked above in ANW's post.
Mixture has unfortunately confused matters. Authorisation is not the same as notice and this argument is what has allowed the police to misquote the Act. Authorisation implies something that cannot be done without consent. This is not the case with private GA flights under the 2000 Act. Notice means exactly that. The police have no power to demand positive authorisation before a flight can take place. They have power to prevent a particular flight if grounds exist under the Act, but that is different from what is being suggested here.
The Cheshire Constabulary letter is a mis-statement of the law in so far as it relates to non commercial flights.
The police have powert to request information about a flight. The information they can demand is in the Regulation which I quoted in the linked thread. It only applies to flights to or from the CTA and not to flights within Great Britain.