You're right, ordinary Fair Dealing applies, the Marrakesh Protocol too I expect (though probably CDPA 1988 has been amended to include those provisions now?).

What I was mainly getting at is that we don't have any general Fair Use exception that is interpreted as allowing archiving (universally) nor retaining copies for private use.

My understanding might be a bit out of date. There were some further rights granted for private use (eg format shifting), not they were quashed about 10 years ago and I'm not up to date since then.