This is a very clear distinction in the UK. There are a number of constraints on someone's rights to make loud noises in the UK, of which probably Section 5(1) of the 1986 Public Order Act is most often used -- it states

"(1) A person is guilty of an offence if he/she:
(a) uses threatening (or abusive) words or behaviour, or disorderly behaviour, or
(b) displays any writing, sign or other visible representation which is threatening (or abusive)"

with all of the usual lawyerly finness about what (b) means being defined by common law precedent and the right to freedom of expression largely being worked out by the courts together with reference to the ECHR. Note that things like animal rights protestors making loud noises at university graduations (where the graduates have nothing to do with animal experiments) have successfully been challenged under this statute – the usual remedy is a high court injunction relocating them to somewhere visible and nearby, but out of earshot, of the thing they are protesting against.

It is worth noting that a long-running tradition in the UK is that if a policeman disagrees with a piece of legislation and the opportunity arises to challenge it favourably in court, he arrests somebody under the legislation in question in what may be favourable but reasonable circumstances (favourable to the defendant) in order to circumscribe the limitations of the act in question. As a consequence, in many ways, I therefore think that this may also be a challenge by police officers to the validity of the conservative government's latest legislation that aims to ban "disruptive" protests, the Police, Crime, Sentencing and Courts Act 2022, which was largely modified to deal with the (IMO effective) protests of the Insulate Britain campaign who recently brought major motorways to a crawl. It is a deeply divisive and very authoritarian bill, and would cover this situation. Specifically, it provides that:

"[...] police forces are allowed to place restrictions on protests they believe would otherwise constitute an existing offence of public nuisance, including imposing starting and finishing times and noise limits, and be able to consider actions by one individual as protests under provisions of the Act. Protestors disobeying such instructions from the police may be committing a criminal offence."

In my opinion, this has already been used to stifle what I find to be legitimate protest – "on 28 June 2022, the day the act came into force, anti-Brexit activist Steve Bray had his amplification equipment seized by police under the 2011 Police Reform Social Responsibility Act. The 2022 act extends the area around the House of Commons in which protest is restricted under the 2011 act." [1].

As recognised by the police, positing the possibility of transitioning to a republic (from a constitutional monarchy) is almost certainly a clear-cut case of a legitimate piece of protest, and so if someone _did_ protest accordingly _without_ being rude and disruptive I think this would be a perfect opportunity for the courts to throw out the charge.

Edit: Further bits of fun law – "public nuisance" is both a tort and a very old legal concept, recommended effectively for removal and updating from the criminal statutes by the Law Commission in 2015 [2]. It is described quite formally in many words (c.f. [2]) but more usefully defined by the fact that it

typically consists either of an environmental nuisance, such as carrying on
works producing excessive noise or smells, or of offensive or dangerous behaviour
in public, such as noisy parties and hanging from bridges.
It also includes obstructing the public highway, though now this is also a statutory offence.

Public nuisance therefore is a much lower bar to offence than either the public order act or breach of the peace would be; and unlike those acts there is no requirement for a mens rea: there is no requirement that the defendant intended or was reckless about whether his conduct caused
the relevant kind of harm. This is quite a key point as it de facto lets the police decide what the line is, with no input required on the part of the defendant. I would be very interested to see a reconciliation of these rights with that of freedom to expression and the broader constitutional right to protest in the UK.

[1] https://en.wikipedia.org/wiki/Police,_Crime,_Sentencing_and_...

[2] https://assets.publishing.service.gov.uk/government/uploads/...