NfkDumpling said 'Would there be a problem if a similar campaign was aimed at “Anyone with a Prostate”?'
Yes, there would. Why do you suppose there doesn't seem to be one?
OldMeg said 'Do we always have to target extremes?'
The CRUK has said it's reason for using that wording was to include transmen. Transmen are a tiny minority of women, who are probably well aware of having a female (i.e. woman's) anatomy. Why you think we need to target them in this way?
NfkDumpling also said 'I rather think [....] any woman with low reading skills wouldn’t read it anyway, even if it said Woman with a Cervix. Would they be able to read Cervix?'
It wouldn't need to say "Women with a cervix", the word "Women" would do, as it always has in the past.
As NfkDumpling so nicely put it, women with low reading skills, or a poor command of English, may not understand or pay attention to the word "cervix". However they may well spot the word "women".