From the 2011 census about 14% of people living in the UK were born abroad. This, of course, is almost certainly an underestimate as it is likely that far more people not entitled to be here failed to take part in the census than those entitled to be here. Mere speculation, I know, but fairly well founded.
Of course the number of people currently here that were born abroad, whilst one measure of the scale of inward migration, tells only part of the story. For a full picture of how recent mass migration has changed the demographics of the UK you also need to look at the numbers born here but to mothers born abroad.
Just as a taster, in 2012 (the latest full year figures I can find) in England and Wales, 25.9% of all births were to mothers born abroad. For those who believe that there are no areas of the UK which are dominated by foreigners, in the London Borough of Newham more than 76% of all births were to mothers born abroad. To further demonstrate the rapid rise in this phenomenon, as recently as 1990 the percentage of births to foreign mothers was just over 10% so in about a generation this percentage has more than doubled and will no doubt continue to increase because (a) immigration shows no sign of abating (quite the reverse in fact) and (b) foreign born mothers tend to demonstrate a higher fertility rate that those born in the UK.
Unlike the Swiss, people in the UK have had not say on whether they prefer these enormous increases or not and successive UK governments have sat on their hands whilst the problems (and it is a problem for a number of reasons) simply increases on their watch.