Although HM Queen Elizabeth II has no actual power, her reign is a symbol of British (or English) culture as well as hospitality. It does not matter who the PM is because it is not the PM who helps create the identity of Britain; it is the Queen. And the Queen is doing just marvellously.
In our day, many young citizens are forgetting their country's history, culture, beauty, and logic. Such absurd questions such as whether the UK should leave the EU or this one, should not ever be taken into serious account.
So many things would change if the United Kingdom were to lose it's monarchy. The name itself for example. You cannot be a "Kingdom" without having a monarch. So what would this theoretic country be named? Great Britain? No because Northern Ireland is missing. Great Britain and Northern Ireland? Too long, and there wouldn't be a nice sounding abbreviation like the UK. British Isles? No because Ireland isn't part of the Union. United States of Great Britain? You see what I'm saying?
You cannot simply detach the monarchy from British culture. The monarchy is part of the backbone.
I'm not British therefore I hope I am not biased. As an American looking in, it seems to me that the monarchy serves a few very important purposes for Britain. First of all, as symbols of the government, they are able to serve as diplomats and representatives of Britain which allows the Prime Minister to concentrate on the actual and important job of governing. The British constitutional monarchy is also very old and historically significant. Britain is the first country ever to become a constuitional monarchy via the Magna Carta. The monarchy has also served as important symbols during times of crisis such as during WWII. Lastly, the monarchy is a huge tourist attraction in Britain and is one of the fascinating things about that country that draws millions of visitors each year. I know many Brits think the monarchy costs too much, etc, but because of the reasons I previously stated, I think the financial cost is worth it. Just my American opinion.
The number one reason for this is simple: the preservation of parliamentary democracy and responsible government. Without the monarchy, there is always a chance that the 'American' system of government could creep in, and that would be a disaster.
The British form of government is the best in the world.
The monarchy prevents someone doing a Hitler, or Stalin style takeover, as the armed forces swear loyalty to the queen, and not the corruptible government. If someone tried to do copy Hitler and do his enabling act, the queen would be able to stop them. It makes us more politically stable.
I am an advocate in self-determination. If the people, by referendum, decide that they want to stay a monarchy (currently 66% are in favour) then who are we to say no? If, however, the people vote to abolish, then a governor-general should be put in place until another referendum would determine form of government. By the way, if Lizzie is given the post of president of the Commonwealth of Nations, she would still have the symbolic stature as she receives in other commonwealth countries, such as Canada and Australia. The people directly, or the prime minister could elect/appoint the governor-general as in Canada, or he/she could be chosen by lottery from amongst those who achieved great honour within the country (Order of Canada or Medal of Honor equivalents) or just any ol' joe/jane.
England's monarchy is today nothing more than a celebrity festival, with the Queen being more of an image than a decision maker. Royal weddings, spending and security costs are expensive to the tax payer, and Buckingham Palace, which ought to be a great tourist attraction, remains closed to the public because of it's VIP host. Invaluable pieces of art that hide under it's camera-filled halls remain hidden from public eye, wasting an incredible chance for education and arts appreciation. Removing the monarch would not affect tourism, as the castles and important landmarks make part of a history of monarchy; tourists don't go to England to have tea with the Queen or King, and so his or her removal will not affect tourism. Prime ministers being the actual decision-makers, making a reform would suit everyone and not be as different of a government than it is today...Except it'd be less expensive. We would also remove hereditary succession, which can lead to obvious political problems if people don't have a say in who is the next king or queen, and save ourselves the trouble of worrying about a particularly bad new ruler.
All monarchies are in decline. It started with the French Revolution and continued with a revolution in Russia, death of Fidel Castro, Mexican Revolution, fall of the Ottoman Empire, German Empire, and Austria-Hungary. This would just seal the deal. End the monarchy and House of Lords. Turn the Head of State of Canada, Australia, and New Zealand to the PM's.