I disagree slightly with mummy. Your children will have a bit of adjusting to do after being in an International School. Slightly different norms and after being encouraged by all those American and Aussie families to be very extrovert my DDs, who were 7 and 11 found it difficult to adjust at first. My older DD found other expats in her class at secondary school making the same adjustments (still her lifelong friend, they are just about to get a flat together aged 21), and everyone was of course making new friends but my younger DD had a difficult time from the existing pupils who branded her wild and weird. Luckily her class teacher had taught in an International School and as she put it the other girls benefited as much from broadening their minds via my DDs experiences as she would have done by conforming . It will help of you can find a primary that is supportive and prepared to understand that challenge and preferably value your DCs experiences. We found the state schools we visited quite disinterested, probably because they knew we had little chance of getting in as late applicants......
However it would help to be in a feeder school (either through a formal feeder mechanism, some schools get preference at particular senior schools, or because you will be in catchment) for a good secondary school.
Popular Catholic Schools tend to be oversubscribed by families who meet strict faith criteria which give them preference. Some C of E schools are similar in spite of instructions from the top to be more inclusive
Grammar School would be a whole different challenge. There are many threads on here about the strategic choices you are faced with in preparing for selective school admissions. Some go in for years of tutoring although the evidence this is needed is highly subjective. It will all be a culture shock after an International School, though perhaps not if you had experience of the selective local Beijing schools
Personally I would go for somewhere known for good comprehensive schools, like Winchester. Sorry that perception is not based on detailed knowledge, just my impression.