The British are relaxed about many things and names are increasingly one of them, but the exceptions are exactly where visitors come to grief. Here is the territory, mapped.
First names, mostly
In modern Britain, first names are the default almost everywhere: offices, dinner parties, emails between people who have met once. The safe rule in formal settings is to begin with the title and surname, Mr Whitmore or Mrs Chen, and let the other person say "please, call me James." They almost always will, usually within the first minute, and the invitation is a small ceremony worth letting them perform.
Mr, Mrs, Ms and Dr
Mr for men, Ms for women unless you know they prefer Mrs or Miss, and Dr for anyone who has earned it, medically or academically. Britons with doctorates are politely proud of them; using the title costs nothing and pleases enormously. Professors likewise. If in doubt, promote rather than demote: nobody was ever offended at being taken for a professor.
Sir Elton, not Sir John
Knights and dames take the title with their first name: Sir Elton, never Sir John, when addressing Sir Elton John. Getting this backwards is the classic error and the one most enjoyed by onlookers. Peers are Lord or Lady followed by the surname or place: Lord Sugar, Lady Bracknell. Should you find yourself unexpectedly presented to royalty, "Your Majesty" or "Your Royal Highness" first, then "Sir" or "Ma'am" (rhyming with jam), and carry on as calmly as you can.
"How do you do?"
The formal greeting "How do you do?" is not a question about your health, your journey, or anything else. It is a password, and the answer is the same password back: "How do you do?" Answering with medical detail marks you out instantly. In everyday settings "Pleased to meet you" or a simple "Hello" does perfectly.
On paper
Letters still follow the old pairing, and getting it right quietly impresses. A letter beginning "Dear Mr Whitmore" ends "Yours sincerely". A letter beginning "Dear Sir or Madam" ends "Yours faithfully". Sincerely for the named, faithfully for the unknown. Emails are looser, but "Kind regards" will never let you down, and "Best" is now the standard between colleagues.
Forms of address are one small chapter of what the Academy teaches. The rest, from introductions to thank-you notes, lives in the English & Etiquette lessons. Send an enquiry to begin.