I think this is a logical solution if you are a writer of such a leaflet, and you know that the leaflet will be read by gender critical people and gender ideologists, and your above-all-else concern is that everyone who needs a smear test should be addressed and told to get one. Two different interpretations:
GC: women and people with a cervix: only "women" is necessary, and is sufficient; "people with a cervix" is redundant (and perhaps offensive, because women aren't body parts, and we don't need reminding that if our cervix has been removed e.g. as part of major surgery, we no longer need to have it smeared)
GI: only "people with a cervix" is necessary, and is sufficient; "women" is redundant (and perhaps offensive, because there are some "women" in their terminology who have to remember they don't need a smear test, and may encounter embarrassment if they fail on this)
By contrast, if they went for "women" only, some GIs who need smears would feel excluded and out of peevitude might not attend (you might feel, and I might agree, that that's their lookout, but the NHS would still pay to treat their advanced cervical cancer); if they went for "people with a cervix" only, some GCs would feel peeved, probably not enough to not attend on the biased assumption that we're reasonable people who know where our interests lie, but more seriously, some people who aren't sure what a cervix is would be excluded.
I don't envy whoever worded this leaflet their job, but I think they did ok.