Can a non-lawyer send a cease and desist letter, or do all cease and desist letters have to be sent by solicitors?
Certainly in English law there is no prohibition on a non-lawyer sending a cease and desist letter.
However, a letter from a lawyer may be much more effective. First, a good lawyer will be able to clearly state the legal cause of action that grounds the demand. Second, instructing a lawyer indicates that the potential claimant is serious. If the potential claimant is not prepared to spend a few hundred pounds on preliminary advice and an initial letter, then in most cases she or he will not be prepared to spend many thousands of pounds on legal proceedings.
Also, there may be specific legal issues to be dealt with (e.g. defamation issues, or in UK trade mark law you need to avoid or at least consider avoiding making a "threat").