If I am not mistaken, religion plays a far more important role in British schools than it does in American schools. I don't think it is a matter of being better or worse, but of each country having a system that works best within their cultural context. The United States is a very sectarian nation, with a wide variety of Christian denominations. Add to that very significant Muslim, Jewish, Hindu, and other religious minorities, not to mention atheists, and you can see why religious instruction would be problematic in the USA even without the First Amendment. 

In the US, the level of religious education offered depends on the school and particularly the type of school a child attends. For instance, public schools must follow the separation of church and state. Students might learn small bits of information about various religions as it pertains to literature or history, but teachers are not to teach or express an opinion beyond the basics. However, a private school is another story. Many parents who wish their child to have a particular religious view point will choose to send their child to a private, religious school. Here children are educated about that religion and more. At a private school, every aspect of that child's education can be looked at from religious perspective. For instance, many private religious schools will not teach the theory of evolution. 

Public schools in the US really do not have a serious religious curriculum of any sort.  In the US, the idea of a separation of church and state is strong enough that religion (in the sense of religious morality) is not taught.  When religion is touched upon, it is only to look at how various religions differ from one another or, for example, how religion affected history.  Thus, there really is not much place for religion in the US curriculum.

While it is true that schools have a heavy influence on children, don’t discount the influence of the church.  Churches have their own religious education programs, but each church is different.  Each church is going to focus on different aspects of religious doctrine and moral guidance.  It is impossible for most schools to be so specific, especially in America where we have separation of church and state, so schools cannot teach religion. 

Parents do have the option to choose church-run schools, which are often (but not always) specific to a particular church’s beliefs.  Parents can also choose blanket Christian schools, for example, that are careful not to choose a specific version of Christianity. 

So parents can look for the church that most specifically matches their beliefs, and if they choose and have the financial means they can also send their child to a school run by that church.