It's pretty unusual to go straight from bachelors to PhD. The only person I knew who did it, managed it because he did a funded summer research project with an academic between 2nd and 3rd years, then did his BEng project with the same academic in third year, then continued that project in a PhD after being encouraged to apply by that academic. I think the funding came from a fellowship that particular academic had been awarded that year as well so, it was really up to her how she spent that and she was obviously happy with that particular student as she already had an established supervisor-student relationship with them and was familiar with his research work.
Without some kind of extended connection with the would-be supervisor (who ideally also has funding for you already), or just being an outstanding student working well beyond bachelors level already anyway, I don't think that's a reasonable goal. This is for BA/BEng/BSc type courses; for MEng/MSci etc integrated undergraduate masters courses it's more common. I gather there are also integrated doctoral programes where you earn an MRes in the first year then continue to the PhD, for which "just" a bachelors may be acceptable.
In any case I don't really think it's that great a leap between GCSE and A-level/equivalent generally. Maybe in languages the gap is bigger (I didn't continue from GCSE language to IB, I did an ab initio language)? The only subject I did where I felt there was a notable gap was art, and I think that was more because the style of the course and expectations from the students were quite different (where we were expected to be working much more independently on self-driven projects rather than creating potted works for preset projects given to us by the teacher).
In a similar vein to that particular example I think the main leap between 6th form level study and uni level study is more about how you're expected to study and learn (much more independently than in A-levels/equivalent) than in the content being necessarily significantly conceptually harder (I mean it will be a step up but not a huge leap I think at first...but there is a big leap in how you're expected to learn that content). The other exceptions are for courses which are very dissimilar to the associated A-level subject (e.g. maths degrees, which are totally different in style than A-level Maths).