Wrong.
Windrush and immigrants from elsewhere came here for two main reasons. Firstly, of their own accord, because there was no system in place to stop them. There was no definition of a British citizen until this was corrected in the 1960s. Secondly, because the governments of their own countries, like in the Carribean, requested our help to deal with unemployed and increasingly troublesome young men.
We did not need labour. As soon as large numbers of troops started returning from Europe, by the late 1940s, there were areas of the country that had an unemployment problem. Adding useless, low skill immigrants into the equation didn't help anybody and understandably caused resentment. As is the case now, wasted scarce resources and drove down wages.
You don't have to believe everything the TV tells you...