It won't.
The US border is now locked down far tighter than it ever was when I was a kid, and the cries for locking it down even more and violently apprehending suspected violators are at a fever pitch. The UK too - like many countries in the recent rightward lurch - has gone from a country where I can just show up to visit to one when were I need to request permission beforehand.
It sure seems like the "concerns about immigration" in the UK mirror those in the US, which in my analysis is a reaction towards the loss of white privilege combined with the loss of economic power. Putting stricter id checks may assuage abstract xenophobia, but the concrete details don't fundamentally change the concrete details.
It's not like Brexit fixed those concerns about immigration.
