Why does that seem important? The “in mice” one makes sense (such results might not be the same in humans), but is there a reason to think that results achieved here in England couldn’t happen elsewhere?

Basically it's definitely possible that you could get cervical cancer without HPV. There's a scientific disagreement about whether in some populations that's say 5% prevalent while others it's more like 0.1% prevalent, or whether they're more similar, but it clearly can happen. Outside science this can turn into "England is very, very white, so, this is basically only a study of white women" which as you observe isn't much like the mouse studies - humans just aren't that different. There are differences, but they're not big enough to suggest this study might be a "glitch" with no relevance for your population like a mouse study could be.

This didn't even occur to me. The title in the submission is cut short; most outlets including this article includes a "close to zero", "near zero" at the end, which has this overly cheery feeling (at least to me) that we can pat ourselves on the back, it's not just done but done-done, on to the next disease, huzzah. Whereas this whole thing is the result of a systematic nation-wide vaccination program that has been going on for quite a while (2008) and a study looked into its effectiveness and found that yes, this concerted campaign may have moved the needle on that population-level gauge.

It can happen elsewhere but it hasn't happened yet! Or who knows!