First a word about different terminology, to avoid any confusion.
What you call 'public' school in US is called state school over here. A 'public school' here is a special kind of fee-paying private school, with history and prestige like Eton, Harrow and Winchester. Such public schools take pupils from 11 (or 13) - 18, and private schools that prepare pupils for entry into public schools are called prep school (short for preparatory), and for younger children, pre-prep school.
So your choice in UK is between (fee-paying) private schools and state schools. Some state primary schools are funded differently - like voluntary aided schools (mainly church-run schools), voluntary controlled schools (church schools but with more state control) and foundation schools (formerly grant-aided schools, more independence) as well as community schools (most common type). They can differ in how they choose pupils, and in some aided schools, they can favour children from certain religious affiliation. Catchment areas exist in some areas (called London Boroughs) and for some schools but not in others. Many parents in London opt to send their children to private schools, partly because of (perceived) better standards, but also because they cannot get them into oversubscribed good state schools and don't want to send them to state schools they consider to give inferior education.
About your children's ages. Compulsory education starts in September following a child's 5th birthday, and they start in Year 1. But most children in most areas (things differ somewhat depending on where you live), children start Reception class in September following their 4th birthday, even though it isn't compulsory (but entirely state-funded, so no fees etc to pay). So, if your older child was 6 before 1st Sep this year, he/she would be in Year 2, and in Year 1 if after. Your younger child would be in Reception when turning 4 before 1st Sep, and in Nursery if after (but again this differs somewhat depending on areas).
As for your choice of schools, to give yourself the best chance of getting a school place of your choice, for Reception and Y1 entry, application should be made around October/November of the year before they start in September. If, as it seems probable, your children will be joining other years or in mid term, or you've missed the deadline, you just have to phone up schools and find out if they have (casual) vacancies. Chances are, though, in most areas, so-called 'good' schools are likely to be full, and the nearest school with a place may be some distance away, or with lower attainment. So it may well be you have to look for private education for one or the both of your children. Again contact individual schools and ask for details. There are inspection reports available for all schools - Ofsted for state schools
Inspection Reports - Ofsted and Independent Schools Inspectorate
ISI Independent Schools Inspectorate for private, fee-paying schools.
There are also (private) American schools that follow US curriculums.