It is good to understand the legal aspects, but you posted a very "lawyer" reply. It's a bit like in business, Accountants take what the Lawyers say and give commercial advice. If people listened to the Lawyers all of the time they wouldn't leave their home.
The best thing to do in this case is exactly what 5UB has done. Maybe I would have changed the locks sooner .....
Well, quite so.
That is why I opened my initial reply specifying that it was in relation to some advice as to what was, and I quote, "legally" done. I also carefully said I wasn't advising what to do, but what the legal risks of it would be, in the face of previous assertions.
Discovering that some unknown person or persons has free access to your home would be enough to upset, and seriously worry, just about anybody, including me. Deciding that addressing that risk, and ignoring potential legal issues, is the sensible and pragmatic approach is also probably what I'd do. My sole point was to correct some previous distinctly incorrect legal advice so that in the event 5UB changed locks, or whatever, he did so knowing the downside. It would take an especially hard-nosed landlord to take umbrage in the circumstances but, as I suspect many here would agree, not all landlords are exactly liberal when it comes to a tenant's point of view.
Now that progress seems to be happening, I'll add that if I were in his shoes, as described, I'd do much as he has done, but probably a bit faster. My view is that protection of my property, and for that matter, my personal security in my own home, is sufficiently important that addressing that security issue is imperative, and urgent, and that the agent getting their ass in gear was mandatory, in my view. If they didn't, I'd deal with the security issue and be quite prepared to sort out any legal concerns, and in the unlikely event of legal action, that too, at a later point.
Personally, the only circumstances I'd want anyone else in my home in my absence, without my explicit permission, would be in an emergency. Other than that, whether for gas inspection, maintenance, or whatever, I would insist on being there. I don't rent, and haven't for a very long time. When I last did, it was as a student and there wasn't really any personal possessions worth worrying about. But as a house owner, if work is being done on the house, I will ensure either I or someone I trust (close family) are present at all times, even if that means taking a week off while windows are being replaced, etc.
Whether I rent or own, my home is my home, and my expectations would be the same either way. An unknown individual, or individuals, wandering around unsupervised, and unannounced, is a non-starter. I would, therefore, for "legal" reasons, inform the landlord, in writing, that any inspections, maintenance etc was only to be done in my presence and that permission to enter unattended is expressly refused. That eliminates the "course of dealing" argument for him.
One concern would be insurance. Policies often preclude cover if entry is gained without visible signs of forced access. This is one reason, though by far not the only one, I would advise landlords to never enter other than preferably in the tenants presence, or at the very least, with explicit permission. Another is the risk of catching a tenant especially of the opposite gender, undressed. Worse yet, catching their kids undressed. That sort of situation opens some nightmare risk scenarios for landlords.
So yeah, I'd either want an explanation, in short order or locks changed. And if it wasn't done as a priority, I'd get it done myself.
And 5UB .... as a longer term security measure, I'd advocate installing a hidden camera or three. If you aren't 100% sure the landlord/agent doesn't have a key, those cameras might be an eye-opener. Had the door not been locked in a different way, would you have known someone had been in? Could they have been in and out regularly before, but not slipped up with how they locked the door? What assurances do you have for the future?
You know the type of camera .... motion-activated, hidden in a book, power block, alarm clock, fluffy toy, wall click, or whatever, recording to micro SD card, etc, or for more sophisticated examples, wifi'ing to your broadband, uploading or streaming to the web, and monitored remotely from your phone It's worth it for peace of mind, and you never know what might come to light.