I'm not sure if it's what was used here, but a lot of areas have some kind of generic "nuisance" law, which basically serves as a general purpose "someone is doing something obnoxious that affects us and we want to provide law enforcement with a way to make them stop" tool.

kagis

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

Under the common law, persons in possession of real property (land owners, lease holders etc.) are entitled to the quiet enjoyment of their lands. However this doesn't include visitors or those who aren't considered to have an interest in the land. If a neighbour interferes with that quiet enjoyment, either by creating smells, sounds, pollution or any other hazard that extends past the boundaries of the property, the affected party may make a claim in nuisance.

Legally, the term nuisance is traditionally used in three ways:

• to describe an activity or condition that is harmful or annoying to others (e.g., indecent conduct, a rubbish heap or a smoking chimney)
• to describe the harm caused by the before-mentioned activity or condition (e.g., loud noises or objectionable odors)
• to describe a legal liability that arises from the combination of the two.[2] However, the "interference" was not the result of a neighbor stealing land or trespassing on the land. Instead, it arose from activities taking place on another person's land that affected the enjoyment of that land.[3]

The law of nuisance was created to stop such bothersome activities or conduct when they unreasonably interfered either with the rights of other private landowners (i.e., private nuisance) or with the rights of the general public (i.e., public nuisance)

https://en.wikipedia.org/wiki/Nuisance_in_English_law#Public_nuisance

EDIT: Okay, found a news article that mentions what they're being investigated for:

https://www.nbcnews.com/politics/donald-trump/four-arrested-uk-projecting-photos-trump-epstein-windsor-castle-rcna231804

Thames Valley Police said in a statement Tuesday night that they arrested four adults “on suspicion of malicious communications following a public stunt in Windsor.” The police added they will conduct an investigation into the incident, and that all four people arrested remain in custody.

Probably this law, though it doesn't sound to me, on the face of it, like it'd qualify:

Malicious Communications Act 1988

It addresses communications "in electronic form", but I don't think that in the everyday sense of the word, a projection would count.

EDIT2: I also wouldn't be terribly surprised if they don't wind up with this actually going anywhere, and just wanted some sort of legal rationale to make them stop it for the moment.

Hurting bourgeois feelings is a crime now I guess

Apparently Malicious communications.

So basically the officers knew full well they have 0 chance of a successful trial. As no way any court will consider the intent to cause distress.

But instead are openly acting to silence inconvenient speech. Having had it made clear by the PA law that the gov wants that.

What is the charge? Projecting an image? Projecting a succulent incriminating image?

It's considered vandalism only when it's not a projection of Gail Porter's arse, apparently.

https://en.m.wikipedia.org/wiki/Gail_Porter