alt Hacker News(IANAL - in the US) I think it's worth clarifying that the third-party doctrine is probably what applies here. You used someone else's computer (Google search and the recorded search history, Claude and the conversation history, or cell phone providers and the tower ping records) and you had no expectation of privacy or any sort of confidentiality (e.g. lawyer/spouse/protected medical info).
I understand that other countries handle this differently and might have more privacy restrictions, but this seems to come down to a judge asking a neutral third-party to testify to what they know about a subject and them responding with search history/chat logs/location pings. I guess if you want to do crimes then you need to stop intentionally revealing incriminating evidence to unbound third-parties.
IANAL either, but I don't think it's even that complex. You can have all kinds of legally-compelled data requests come up in a subpoena or in discovery. Even if the models were completely self-hosted on the owner's own computer, the opposing side could ask the court to make you hand over those logs, in the same way they could ask for a copy of your journal. While the 5th amendment right to not incriminate yourself still holds, I think a self-owned AI would be just as subject to a subpoena as your diary or calendar or whatever else. And in no case could I imagine it would get the extra protections of attorney-client privilege.