The UK is not 'full'...It's not blaming foreigners, the fact is we are a small country that is full.
Only about 3% of the UK is built on, or to put it another way less than the land revealed when the tide goes out...
The UK is on paper a rich country, but the money is concentrated in a small percentage of the population hence the perception that too many people are affecting what can be seen easily - namely public services!
So then too many ignorant people are sucked in by the right wing press/politicians, and are persuaded to blame immigration for what are home grown problems.
They also fail to understand that the UK has an ageing population, and thus the worker to unproductive people falls unless there is immigration...
We are now seeing problems in an ever increasing number of occupations because of a lack of employees which has been made worse because of brexit...