The proper way to address the British Prime Minister in a letter is to begin by calling him or her "The Honourable." If he or she is a member of the Privy Council of the United Kingdom, he or she should be addressed as "The Right Honourable."
When writing the address of the Prime Minister, it should include "The Honourable" followed by Prime Minister and his or her name. The next line should note the Prime Minister, followed by a line noting the Office of the Prime Minister. The address should follow after that.
Within the letter, the salutation should be noted to "Madam or Sir" followed by Prime Minister and his or her name.