doesn’t the UK not have a constitution because it’s basically whatever the Monarch says?

No. In the 2024 British system of government, the monarch has essentially no power. The upper house of the bicameral legislature, the House of Lords, has very little power.

Virtually all power is wielded by whoever controls a simple majority in the House of Commons, the lower house. They can do anything (short of limiting what future Parliaments can do). Rewrite any law, whatever.

My understanding is that this is largely a result of having a system constructed in an aristocratic period, and shifts in power occurring, but the system of government not being restructured.

In, say, the US, the constitution limits and specifies the powers of government. It places a set of constraints on what Congress and other bodies can do.

The British system evolved from a situation where the aristocrats were represented in the House of Lords, the monarch was his own thing, and the rest of the public in the House of Commons.

Part of the transition over time was to limit the power that the king had. The Magna Carta constrains what the king can do, shifted away from an absolute monarchy. In that sense, the UK is nominally a constitutional monarchy.

But since the king has no real power today, power has shifted from the monarch, the written restrictions on his power are essentially meaningless.

Over time, the aristocracy also lost power. The House of Lords lost most powers it has, and today has very little actual power -- I believe that perhaps the most-notable is the ability to delay legislation for a period of time.

Where basically all the power has concentrated is in the House of Commons.

And that has no real restriction on it. The Magna Carta doesn't restrict Parliament. Parliament has modified text from the Magna Carta with a simple majority before.

There is no power of judicial review over the legislature on the UK -- laws Parliament passes cannot be ruled unconstitutional. The executive is subject to judicial review -- there were some notable UK Supreme Court cases in the part few years relating to the actions of the prime minister. But the legislature is not -- the judiciary cannot rule a law passed by Parliament to be unconstitutional.

I once read something calling the UK an "absolute republic". I think that that's probably a much-more-apt description for the state of affairs in 2024 than its official designation as a constitutional monarchy. The UK, as it exists in 2024, isn't run by a monarch whose powers are limited by a constitution. It's run by a simoel majority in the lower house of a legislature who have no real limitations on their powers.

Not only that, but the one great convention is that Parliament cannot be bound. So Parliament cannot go write a constitution and then have it bind future Parliaments. That future Parliament could rewrite it as easily as they could do anything else, with a simple majority.

My belief -- and this is me talking here, not some British constitutional law expert -- is that the plan had been to move the UK to something that looked more like a conventional, constitutional republic by way of its EU membership, by some fancy legal and political footwork. If the UK signs onto a treaty, then it cannot do something against that treaty without violating the treaty. Parliament can still, perfectly legally under UK law, instruct the UK to violate treaties. But that would have consequences with the rest of the EU, and there would come a point in political integration where being in trouble with the EU would be unthinkable, so the UK would have become de facto a constitutional republic (or part of a constitutional republic).

If that was indeed the plan, I'd say that it was actually quite impressive -- the UK has a very elderly system of government that has, over time, managed to transform itself into very different forms, de facto without revolution or an official break with the past system by kinda kludging things, and some elaborate legal reasoning. This would have added another transformation.

But with Brexit, I suppose that that's off the table, at least for some time.

And there is basically an agreement to off on whatever the MPs decide because otherwise they would officially overthrow the monarch

Sort of. There are a lot of things that are formally done by the King by way of the King-in-Parliament, where British sovereignty is theoretically vested, is the "ultimate power" in the legal system, the way the US Constitution is in the US. But in practice, the King doesn't really have a choice as to whether to do them or not, and he'd basically get ignored if he objected, absent some sort of real question as to the legitimacy of Parliament acting (e.g. if there was a dispute over election fraud determining control of the House of Commons, I expect the King's voice might bear weight). It's really the Parliament, and within Parliament, the House of Commons that holds political power today.

Also, one last note on the British system of government, as to your comment:

UK not have a constitution

So, personally, I'd agree with you here. The British don't have a constitution, or at least not one with meaningful effect aside from other British law, aside from maybe the convention that Parliament cannot bind future Parliaments. But that isn't the British legal view of things. Their take is that they have an uncodified constitution, that many different (not always specified) documents and traditions make up their constitution. My take is basically "well, in what functional way does that differ from not having a constitution"? But in the name of completeness, just wanted to keep things correct.

https://en.wikipedia.org/wiki/Constitution_of_the_United_Kingdom

But even if you adopt the British take on this, whether or not something is part of the constitution or just regular law becomes essentially an academic question, because there is no special status that constitutional law holds relative to anything else.

But in Canada, there is a difference between law that is part of the constitution and all other law, so that becomes suddenly a real and meaningful distinction.