I have some knowledge of this (albeit I may be corrected). You will be surprised to learn that on average Catholic schools have an average of 30% of non-catholic pupils in attendance. I think though in London that figure is a little lower because of over-subscribing. The Catholic Church has an agreement to accept non-catholics depending on how close they live to the school.
It's important to note non-catholics can be excused from religious services and religious lessons. Many non-catholic parents choose this and it seems to work well. Many others choose to let their children attend some catholic instruction where it imparts ethical teachings or activities of a charitable nature to the local community. Each school has it's own arrangements. Opt-out non catholics are supervised by a teacher during such times and are expected to do course work, or play chess or maths games. So the old worry of indoctrination is an old wives myth, non catholic parents choose the degree of their chid's involvement.
The most sought after catholic school in our area is Wimbledon College. I think there is an exception there to the usual rules of 30% admission for non-catholics because it was/is? actually owned by a catholic sect, the Jesuits who opened up the school to non fee attendees in the mid 70s. So they had some freedom to negotiate with Merton Council outside of the diocese (most Local Authorities welcome faith schools as a cheap answer to over crowding in their jurisdiction.) Though Wimbledon College too have a % of non-catholics (albeit in a very expensive housing area).
Most catholic schools are based on legal arrangements founded in 1870 when state education began. The catholics, after a long memory of confiscation and seizures up to emancipation 1828 were very careful to legally maintain their independence and tie everything down in covenants, trusts and other legal structures. This means it is quite difficult for HMG to interfere with their existence. In modern times this translates to the church buying their own land and paying for the build. Then the school pays building insurance and maintenance costs. The state pay staffing costs and facilities, e.g. books, to gyms, to heating, etc. Many Catholic schools have a building fund where a each child pays a few pounds each month in term time into the fund (not compulsory but most pay). In several cases the diocese covers building costs. The catholic schools follow the national curriculum and safeguarding standards based on government guidance (KCSiE). They have control over who they employ for teaching (normal state has to follow Dept of Ed guidelines) and the same with free choice of who they appoint as governors. The freedom to choose teachers has the benefit seen in many Independent schools to take on teachers from overseas, Antipodeans and French / Flemish are common among staff which gives some variety.
I hope this gives some useful knowledge to NV readers.