I'd be a geek and use data and say that 87% of children get their first choice primary school place.
The most challenged areas in London the figure is 70%, and there are NO cases of children not being offered education provision within their borough.
I'd go further and say 80% of secondary school children get their first choice too.
The issues with school places have been magnified by the fact that since 2010, when people started warning of a need for increased places, there has been little or no movement by the government to increase the number of places available, magnified by the cutting of places as demand dropped between 2000 and 2008.
Now, immigration is going to be a factor, but only 4 % of the UK population are EU immigrants, and a smaller number than that. If you look at CB claims for 900,000 children ( many of these will be the children of UK nationals too) and there are 8.56 million school children in the UK then a little over 10% of children in UK schools are going to be the child of one EU national, which is a generalisation a good number of these children won't yet be of school age.
The number of children in primary schools increased by 105,000 between jan 2015 and Jan 2016, as was in line with expectations. However the proportion of childen in infant classes has fallen very slightly since last year but is still higher than in previous years.
Over all, the admissions policies to schools (and if you've ever seen any of the loopiness about school "choices" on here and some of the stuff people do) are strict, they don't favour anyone British or non.
What I'd say to someone complaining about immigration is:
"If an immigrant child/child of immigrants has got a place at the school you want it can be down to a myriad of factors, and without that child there it doesn't mean that your child would get the place anyway. "
Its not fair to blame immigration, its a cop out.