IMO this is the hardest to prove. Private Investigators (& investigative journalists) gather information in a variety of ways.
Yes, some, likely illegal, but mostly from really digging deep in a way the public couldn't or wouldn't know about - public databases, records, random websites, tax records, land deeds etc etc, that can give snippets of info that they can use to piece together. Majority of it is gathering info via a systemic double checking, a tiny amount via illegal means.
I think it'll be hard to prove the illegal info gathering.
"They illegally found out which flight I took" - No we checked what flights/airlines flew to specific destinations and sent someone to hang about an airport for days. (as an example).
And I find it slightly laughable that H&M "rejected" the RF. They may have left the UK and working as Senior Royals but they insist on being titled and giving their children titles of Prince/cess. That's hardly rejecting Royalty - that's embracing it.
And Andrew is scum. Zero argument about that.