When did the British monarch become an emperor?

Victoria was the first to have an official title, but I believe Henry II, also known as: Henry FitzEmpress or Henry Plantagenet, was the first to be addressed as one, so around 1160 could be an answer to your question.

Henry ruled as Count of Anjou, Count of Maine, Duke of Normandy, Duke of Aquitaine, Count of Nantes, King of England (1154–89) and Lord of Ireland; at various times, he also controlled Wales, Scotland and Brittany.

The story I've heard is that, when Victoria's daughter married the Czar of Russia, she essentially became an Empress, and therefore 'outranked' her mother. Since that wouldn't do at all, Victoria took the title of 'Empress of India' so she could maintain her status as the head of the family. The source of this scandalous bit of character assassination was a college professor who follows such things far more closely than do I, and who am I to disagree with a college professor? * snerk *

I guess it all depends on what you call an empire. Henry VIII looking for a way to get divorced from Catherine of Aragon so that he could marry Anne Boleyn asked people look for interesting legal solutions.

Thomas Cromwell managed to discover that since the Anglo Saxon times England had always been an empire, thus Henry was an Emperor, ie he did not have take orders from other rulers such as the Pope.

See Scarisbrick, John (2008). Henry VIII. Berkeley, CA: University of California Press. ISBN 9780520011304. page 314.

But this is medieval politics, as has been said in other answers, the current British Empire started in the Victoria era

There is no formal ritual which bestows the title "Emperor". The term doesn't have a legal meaning, there no copyright, nothing to prevent Joshua Norton from declaring himself to be the Emperor of North America. (Pedants may argue that the term derives from Imperator, which does have a formal requirement and bestowal ritual, but since the fall of Rome that title is not relevant.)

Arguably the Monarch of the United Kingdom became an Emperor in 1759 after the Battle of Quebec. Great Britain defeated France and inherited title to French lands in North America. That is the event that my teachers asserted made the UK an Empire, although there isn't really a legalistic justification for the change. The UK suddenly had Imperial problems, which I would loosely define as "How to perform governance in a foreign land where domestic governance obviously doesn't apply?". North American colonies were not "represented" by members of Parliament, nor by the House of Lords (One could argue the Lords Spiritual, but ...) British law applied and British rights, but governance would have to be exercised differently. There could be no Bishops, no Lords, no Barons, no Marquis, etc. The institutions of governance that relied on these officials would have to be restructured. For me, that is an Imperial problem and that is why Britain is suddenly an Empire. (note that Scotland had Scottish titles and Scottish governance institutions that the UK could co-opt; there were no such institutions in North America. While the individuals serving as Governors' may have had titles, they were serving as the King's representative, not as titled Lords.)

Flip the question the other way - What would have happened if Britain were never an Empire? What if there is no point at which one can say "at this point Britain is not an Empire, but at the point it is." Nothing in history would have changed. Emperor is a word we use out of convenience to describe Executive authorities that must incorporate a diversity of governance institutions. (In point of fact, I believe that Adams suggested that G. Washington be addressed as his Imperial Majesty. Thank goodness he was defeated.)

They sometimes used the title "King-Emperor". The main cultural region was England (& Britain), which had historically been "Kingdoms" not "Empire". Hence it was more culturally acceptable for the reigning monarch to be called "King", not "Emperor".

Victoria and her successors were king/queen of the United Kingdom (England + Scotland + Wales + Ireland) and emperor/empress of India in personal union, in part at least so as not to be outdone by the Hohenzollern (king of Prussia and emperor of Germany) and the Habsburgs (emperor of Austria and king of Hungary). “RI” (regina-imperatrix) is the exact equivalent of “k.u.k.” (kaiserlich und königlich).

People talked about the British Empire, but nobody ever actually officially created a British Empire.

There was no monarch of the British Empire, no prime minister of the British Empire, not cabinet of the British empire, no parliament of the British Empire, etc.

There was merely the United kingdom and a bunch of colonies controlled by it with no central institutions except for those which the United Kingdom already had.

Victoria was persuaded to take the title of Empress of India by then prime minister Benjamin Disraeli in 1876

Partly this was to stop the political embarrassment that her daughter was married to the heir to the German emperor which would have made her daughter an empress and so outrank her - thus making Britain less important than Prussia.

Mostly it was that Disraeli was an expert politician and able to twist the monarch and public opinion to his purpose. Giving the head of state an impressive title that made her happy, the people happy and increased the country's international standing - at no cost - was a masterstroke.