there weren't enough men to do the jobs. The reason being, they had died in the war.
A brief search in Google shows that in 1945 the UK population was about 50 million. The casualties of the WW2 are less that half a million, which is a little bit less than 1% of the total pre-war population.
You said that people are not interested in historical reasons about immigration - well, it is too bad. The history is not something boring they teach in school, knowing the history and understanding the reasons that lead to one or another thing could help avoiding costly mistakes now. The immigration in mid 20 century was a legacy of the British Empire - until 1962 anybody from the commonwealth countries could come, work and stay in UK, no permit required.
There are many things that should be changed in the current situation. For example the fact that the asylum seekers can not work for at least an year is ridiculous. If the country has to accept these people, it is madness to put them on benefits.
The consequences for breaking the law should be changed. There was a program on the TV about the work of the border control officers. Well, they find illegal immigrants hidden in lorries, very politely help them get off the lorry, politely ask them if they have any documents (most of them don't - deliberately) and the most they can do is to send them back, again very politely. The unsuccessful immigrants cheerfully go back to catch the next lorry. If they knew that once caught they would go to prison, a real one (which probably don't exist nowadays in UK), not a comfortable room for one person with a TV, internet, etc., maybe they would think twice before trying to cross the border. And IMHO, we care too much about the human rights of the criminals - both British and foreigners.
The whole benefits system should be changed - for everybody in the country - British or foreign nationals. I know people (British) who hardly ever worked in their lives because the system allows them to live quite comfortably on benefits. I don't mind part of my taxes to be used helping people between jobs, or people who can't work because of medical conditions. But I don't feel it is a good cause to help those who could but prefer not to work, be they native British or immigrants. It is a joke to pay such benefits that one could live better on the dole than on minimal wages.
Another lesson that our politicians should have learnt from history is that the level of education available to the citizens is an indicator of the country's success in the future. The rapid economic growth in China goes together with severe competition in schools and other educational institutions. China has the biggest number of engineering graduates, India follows. At the same time we have huge cuts in the education funding. In order to survive the universities try to attract as many foreign students as possible because they pay higher tuition fees. This, and the increase of the tuition fees, lead to reduced numbers of British students. How difficult is to realise that in not so distant future there will be shortage of specialists in the UK and they would come from China, India, and other countries who are wise not to chop off the branch on which they are sitting.
I should become a politician