Hello, thank you for the question.
I assume that your girlfriend's family doesn't know that she's in London. That's pretty obvious.
Sorry, I didn't see your whole post. I'm reading it now.
A welfare check is when the police are alerted that someone might be in jeopardy, like a person who is a shut-in and hasn't been seen in a while, or where there's a court order between two people to have no contact due to domestic violence and a neighbour calls in to say that they are both under the same roof.
The police can do the check at the last known Markham address and will have to report that she doesn't appear to be there. At that point, the family could call her in as a "missing person".
You can't do anything to keep the police from responding to a report about her or a welfare check. If the police track her down, she can remind them that she's an adult and that she simply wants nothing to do with her family.
Regarding private investigators, you can't stop her family from hiring them if they know what city she's in. The police shouldn't tell her family what city she's in, but if they appear to be sympathetic then the police will say she's in safe place. If in fact they know.
You're asking if there's any way to stop her family from calling the authorities. Unfortunately there isn't. But if she's fine then the police will close the books on this welfare check pretty quickly.
Does that answer you? Please reply with question or comment.