I can see that it is becoming a major political issues in many areas - particularly parts of London and other big cities.
DD has been at 2 different primaries - both had over 70% EAL, and the vast majority of white children were Eastern European. The parents tend to be bright, ambitious for their children and interested in their education which can only be a good thing overall, but it does cause problems on a practical level as well as affecting people's psychological views (even if perceptions are sometimes worse than the reality)
The area has had a lot of new building - mainly 2-3 bed flats, but in London it's not unusual to have 2 children and live in a 2 bed flat as anything bigger is unaffordable both to buy or rent and you can forget about social housing.
A new and large block of flats going up near a primary school can effectively wipe out the chances of people who live in larger houses further away due to concentration of people on a small footprint. The majority of larger properties here are social housing and are occupied by families who have lived in the area for generations.
There are also 2 very good RC schools which used to take a mix of children of RC parents and a good number of children whose parents were of other faiths or none. A lot of Eastern Europeans are RC and practising and there are now no places for unbaptized children. They also qualify for the 2 local CofE schools by ticking the 'children of other faiths' box giving priority over distance applicants. Long-term locals are fuming - understandable even if you have no issues with immigration.
They can't get their children into the schools where all their friends and relations children have gone and having to travel to schools further away now makes it difficult for granny to pick the kids up and the children now don't have local school friends.
When I was a councillor, by the end of my term, the biggest number of complaints I got from constituents was about lack of school places due to recent immigration or from 'posh people' buying the new flats and being unable to get appointments with the local GPs. When I was first elected, the big issues were housing and transport.
I have no idea what the answer is - even if the money was there, there's no physical space to build new schools round here, most existing ones (bar the faith schools) now have 2/3 form entry and are now taking bulge classes on top.
Perhaps building 5-6+ storey high-rise schools and retaining the postage stamp of outside space is an option.