I'm assuming you're talking about primary, as all children who have taken First Holy Communion irrespective of whether they were baptised in an RC church or another trinitarian denomination are Catholic under Canon Law.
If they haven't arranged a Baptism until just before school application time, that's unfortunate. Mainly because, as is required of and believed by Catholics, their child has therefore not benefitted from what is perceived to be the protection of their soul. Can't imagine that there are priests who would say 'ah, don't worry about it, you can do it just before the school applications are due'.
Attendance for a priest reference, well, it's not as though there are only Masses on Sundays.
Children with the school named on an EHCP get a place irrespective of their faith and go to the top, no application necessary, it's just an order. That's the law.
The law then states that looked after child or children adopted from care are given first priority. It's legal for children who are catholic or in the care of a catholic family to have the very highest ranking and for other looked after or adopted from care children to be ranked below catholic applicants.
There's often an exceptional circumstances category. It often forms the main part of discussions in Admissions Committee meetings - is there evidence of the circumstances, is it likely on the balance of probabilities that the circumstances are as presented, is it enough to give them priority over all other applicants, is it likely to be successful upon appeal (so it's less stressful for the applicant to just do it right first time), is it the right thing to do? This can be things like having a terminally ill parent, being a refugee, being very unwell themselves but not with an EHCP, emergency moves fleeing violence - you know, actual exceptional circumstances. They usually do not have to be catholic.
After that, it's usually siblings and then all other catholic applicants. Catholic applicants can be further placed in order on the basis of attendance at Mass, when they were baptised. Not all places do this, though. But not all places are hugely oversubscribed.
Teachers' children can't be subject to the Catholicity requirement, as it's not legal to refuse a job to a non Catholic (beyond certain SLT positions) and that would create a two tier staff situation. If staff children are not in the published oversubscription criteria, they are just dealt with as any other applicant.
After that, it's people in the process of becoming catholic, other faiths, then no faith or no faith evidenced.
Each applicant is ranked within their category on the basis of distance. It's the only legal tiebreaker and the distance is usually one provided by the local authority. So the PP who didn't get a place was likely because they lived further away, assuming they were also of the same faith and all other things were equal. It's highly unlikely to be due to racism - if nothing else, if anybody were to attempt to remove non white observant catholics from applications in the majority of urban areas, they'd find themselves undercapacity in September. And fired immediately afterwards.
Catholic schools tend to be more diverse in terms of both ethnicity and income level. Their kids come from further away, which reduces the effect of the school being in an expensive area - in some ways, it's a leveller that a non faith school in an expensive area doesn't have - and it's quite possible that there are a larger number of Muslim families are able to get a place in that school, depending upon how oversubscribed they are, which means that poorer families of other faiths also have a good chance of not having to go to the nearest LA school, the one that's undersubscribed and may have a poorer Ofsted rating.
Whether this is right or wrong is moot; but it is what is legal. Any stories about governors getting their friends in, having to do the flowers, make donations to the church or school or staff pulling strings is bollocks. Utter bollocks.