AN: Unlike other parts of this collection this chapter isn't going to be in-universe and more of an essay on some particularities. I apologize if I ramble in it.
Primer: Wizengamot and Ministry of Magic
I wanted to explain a bit more about the political system of Magical Britain, but it's not going to be relevant for years to come so I didn't want to break up the story with a page long essay. It's also not really secret knowledge or anything and some of you might appreciate the more in-depth information. Bear in mind that details might be subject to change.
I've taken inspiration from several different sources, most notably the Holy See, Plato's earlier works on politics, the Roman Republic, and the British Empire. Keep in mind though that magical nations aren't the same as ours and have to deal with different challenges. They are all considerably smaller, both more and less diverse (race and gender have historically mattered much less than species and non-magical heritage or even culture), generally enjoy a higher standard of living (basic needs are always met as desperate people with a ability to shape reality to their needs are a serious threat to the statute of secrecy), and obviously have to hide from the muggles.
The Wizengamot is part Cardinal Congregation, part House of Lords, part meritocracy. Members mostly serve for life. They either perform an extraordinary feat that benefits society (killing a Dark Lord), are voted in out of a selection of aristocrats and more minor 'heroes' (inventing some helpful medicine, being a highly decorated Auror, donating a lot of money...), or sit on it by virtue of a high position in the ministry. In that case not the person themselves holds the seat but it comes with the job (Amelia Bones, Umbridge). Not everyone can vote for the Wizengamot members that are elected though, as a certain income is required for voting rights being granted (as used to be the case in the muggle world) and there is no anonymity. That's partially out of necessity as it prevents tampering with the votes if everyone can check if their vote was registered correctly, though it's certainly not an ideal solution. The more I think about it, the less viable anonymous voting is in a magical world like the one in HP. It has several of the problems E-voting has on top of some inherently magical problems like mindcontrol. For that reason voting is restricted to those that theoretically are less susceptible to vote selling and votes are open and reviewable so clandestine tampering through any spells can be caught. This is an example of an unique problem that we don't have.
The Wizengamot's duties are selecting the Minister of Magic and overseeing the Ministry, delivering high justice, and functioning sort of like a mix between a board of directors, Parliament and Supreme Court. Normally they do not write the laws but they have final say over them. There are two regular multi-day meetings each year where laws and appeals are discussed plus a low number of irregular meetings as needed to hold court. The Wizengamot can empower smaller committees to handle less important cases or look into specific laws/dealings of the ministry. The community is small and stable (though one might say stagnant) enough that the judicial as well as legislative workload isn't too great. Serving on the Wizengamot is a honorary office and the members aren't paid.
The Ministry of Magic regulates daily life in the magical world and generally does a pretty decent job at managing Magical Britain while keeping everything hidden from the muggles. In order to do so it has a small amount of direct legislative power and can enact regulations at the discretion of their department heads. Far reaching laws still have to be brought in front of the Wizengamot but for example someone like Arthur Weasley with his middle management position has the power to bar the sale of flying carpets on the grounds of more general laws. His Muggle Protection Act however had to be voted on in the Wizengamot.
The Ministry also has low judicial powers. Minor infractions are judged and punished by the Ministry without involvement of the Wizengamot. Mostly this covers fines, community work or short prison time in the low security ward of Azkaban; Fudge was actually in his legal right to ship Hagrid off to Azkaban although certainly he had no moral right. Technically everything can be appealed, either to people higher in the Ministry or the Wizengamot itself but in practice it's difficult to get a different outcome. The Magical World has no lawyers, the best one can hope for is that someone more knowledgeable speaks for them to the Wizengamot.
The relationship between the Ministry of Magic and the muggle government is similar to that of two sovereign nations that need some limited cooperation with the magical side holding most of the cards. The wizards prefer to deal with Prime Ministers and not Monarchs because the elected politicians change faster and are easier to oust if necessary (which has happened). The Queen might know about the magical world but has no interaction with it. Historically there was certainly some interaction between monarchs and wizards living on the same land, but relationships were rarely more than luke-warm. Wizardkind as a whole most certainly did not swear oaths of fealty/loyalty, although individuals might have done so in the past. Today the Statute of Secrecy would make such a thing illegal. There are no wizards with muggle noble titles and although a case can be made about there being a magical aristocracy it's really more of an oligarchy/plutocracy.
Internationally Magical Britain is a superpower that lords over several dominions, protectorates and vassals, although their subjects generally enjoy greater autonomy than the muggle ones under the British Empire. One of the other main responsibilities of the Wizengamot is to assign liaisons to those colonies and their self-elected governors in turn have to work with the Ministry to regulate trade. Without wars to finance Magical Britain was always satisfied with a minimal share of colonial taxes and although even that certainly doesn't sit particularly well with some of their subjects, it's just not enough to stir up enough anger for the wrong (or right?) kind of Tea Party.
Obviously from a modern, western perspective this is hardly an ideal situation. The missing separation of powers, violation of basic democratic principles like universal suffrage, anonymous voting and equality before the law (technically the magical world has that too, but even more blatantly not in reality) make it easy for the system to become despotic. Certain freedoms are also restricted by design, especially when it comes to interactions with the non-magical world and the Ministry is given a great deal of leeway when upholding the Statute of Secrecy. Counteracting this at least partially is the smaller size of the population and great personal power each wizard and witch potentially holds. It also helps that many wizards subscribe to a black and white morality, which all together makes it difficult for a dictator to stay in power under normal circumstances. Some have tried but rarely managed to do much harm and eventually they all got kicked out. Dark Lords are a separate issue but they too never survive long once in power as historically most of them inspired fierce opposition. Over the past century however there has been a gradual shift that might have undermined those self-protective mechanisms as was seen with the rise of Grindelwald and Voldemort.