Britain's democracy evolved gradually from feudalism, without much in the way of violent uprisings or outright revolution. The entire system is based upon incremental changes to give us the democracy we have today with various checks and balances. The monarchy is a vestigial part of that system. Look at stuff like the Privy Council which continues today, and Magna Carta, which was one of the starting points for it all. Technically, the monarch still has a huge amount of power, but in practice they are required not to exercise it. They have to sign everything into law; in a very real sense the law still is what the monarch signs their name to, as is the freehold system of property rights. All the land is owned by the monarchy; you get a freehold on it.

Like the enlightenment and the industrial revolution, democracy wasn't imposed, it came into being here for the first time, and while it's not perfect, there's a lot of factors, including sentimentality and inertia, keeping the status quo going. You could argue that a clean start with a proper written constitution and federal government (with separate state governments for the different countries) would be a good move. But like for any working system, it's painful to disturb entrenched structures, and there's always the risk of breaking something important. The US was able to do a better job; it's easier when you can start from a clean slate with the knowledge of hindsight.

Also, the monarchy has for the most part been a net positive. Compared with what elected MPs have done to the country, I see it as a rather more benign institution in comparison, in recent centuries anyway.