Having fought together against Hitler provided that sense of unity that is necessary for people to willingly financially support total strangers.
For a welfare state to succeed that sense of an entire society's belonging figuratively to the same family is a conditio sine qua non. Another necessary condition is a general sense of trust, the perception that the welfare recipients will not exploit and take advantage of the welfare donors' generosity.
Both these indispensable conditions have now been lost.
Immigration is undoubtedly a cause of this loss, but it is not the only one. The welfare state over the long decades of its existence has produced what American sociologist Charles Murray and others call the "underclass" (incidentally another sign that the US, contrary to European popular opinion, does have an extensive welfare state too).
The underclass is a new social class, it is no longer the working class. It is not characterised by its economic status so much as by its behaviour, mores and ethos.
It has a disproportionately high illegitimacy rate, school drop-out rate, unemployment rate and crime rate. It is anti-social in its outlook, attitudes, rules and codes.
In the US the underclass is disproportionately black but in Britain it is mainly formed by indigenous Britons.
This is why only solving the problem of immigration will not solve the welfare problem.
That pro-welfare consensus after World War II does not exist any more. Now the opposite consensus exists.
One of the purported reasons for the creation of the welfare state was the desire to reduce income inequality (or relative poverty), portrayed as a cause of social unrest.
I don't know if enough evidence exists that income inequality causes social unrest. As is well known, correlation is not causation.
Social instability is more likely to be caused by poverty - real, absolute poverty in relation to one's needs, not the feeling of envy generated by looking over the fence at the neighbour's garden and seeing there rare orchids not found in one's own pretty but not luxurious garden - not income inequality. Moreover, it is caused by agitprop elements who keep telling people that they are treated unfairly (a bit like the US black leaders, that author Tammy Bruce calls "merchants of misery", who have made a career out of perpetuating in blacks a never-ending sense of victimhood and desire for retribution).
At the moment, much of Western social unrest is caused by the disastrous effect of governments' overspending, mostly due to elephantine welfare states.