How close does my old passport photo have to be to my new passport photo to not require a countersignature?

This appears to be an area where the guidance (possibly deliberately) has got more vague over the years. I'm sure I remember previous versions explicitly stating something like "the effect of ordinary ageing may be disregarded" or something along those lines, but now it looks like the line in your question is all you get. There does appear to be an upload of advice to examiners, which may be relevant, particularly the paragraph

The photograph to be used for the passport must show the same person and, appear to show the person at the same age, as in the certified photo. If the original photograph received is acceptable, but the countersignature is not then, to, cover all eventualities, request 2 further photographs, one countersigned. However, if you are satisfied that the photograph already scanned onto the system shows the same person, and appears to show the person at the same age, as in the new certified photograph then the photograph already scanned onto the system can be used in the passport provided it passes all manual and automatic photograph policy standards.

Not that this provides many more details from the applicants point of view.

Personally, I'd say that if relatively casual acquaintances say they can recognise you from both the old photo and the new ones, you are probably all right to proceed, but if you're particularlly worried, then one possibility, depending on whether you had any friends with the correct professional background in your old location, would be to post the photos back and forth to them, rather than travelling yourself.