If low-skilled immigration is so good for the UK in terms of multiplier effect, then why hasn't it driven up the wages of the low-paid?
An increase in the supply of labour, which happens when there is an increase in immigration, increases the potential output of an economy. Additionally, as the majority of immigrants are of working age they also increase demand in the economy. So, the theory goes, immigration is good for the UK (in an economic sense). However, the greater the supply of labour - the less bargaining power the worker has. Additionally, employers generally set wages, not workers and as a country we have progressively worsened the ways in which workers can collectivise themselves to re-balance the power difference between employers and labour. But at least we have weakened those bad unions.
So what we have is a situation where immigration benefits the economy in terms of what it can achieve and increases demand in the economy meaning more is produced to meet the demand but power over wage levels are ever more in favour of employers who, in pursuit of profits, dividends to shareholders and other factors curtail wages in favour of other "priorities" - thus the value of labour is below what it should be. Consequence, a low wage economy.
There are few, if any, reasons, why wages would be driven up so I am not sure why you are making this point. Before the referendum campaign, these were the issues that led me to believe that we shouldn't be in the EU because the supply of labour was preventing us addressing low pay and pay inequality but it became very clear that nobody on the leave campaign was arguing for a Brexit that addressed any of these issues and it is very clear that the current government has no intention of dealing with these issues...so this is what is going to happen.
We will leave the EU. We will still need low skilled labour. We will rely on immigration for that labour. Mobility is the biggest barrier for low skilled workers so in all likelihood the main source of low skilled labour will continue to be from our near neighbours in Europe and because we will need them to continue coming to the UK...we will let them. This government will do nothing to address low wages through legislation so there will be little difference in wages in the foreseeable future.
We will be out of the single market but still broadly have free movement of labour from EU countries because our economy will continue to depend on it but we will have some notional control on the numbers. Those controls are likely to be employed politically rather than with any economic foresight so will be a mostly destructive tool that will cause needless harm to our economy. Wages won't go up. Unemployment won't go down. The economy will falter.
Alternatively, we can rely on our own citizens to plug the gap vacated by the immigrant workforce. Do not think that suddenly all our unemployed will suddenly pick up the mantle unless you are going to force them to do it. The consequences of forcing people into work that they do not want to do is rarely favourable and I am fairly sure that most businesses would not want such labour forced on them. No, the more likely scenario is that it would be those workers who aspired to become skilled who who would end up taking those jobs...therefore losing the opportunity to become skilled workers. None of the reasons why these jobs are low paid now would change, so they would continue to be low paid. We would then transfer the deficit in our labour market to the skilled workforce and we would require immigrants to plug the gap. Now these people have a skill, we can't just pay them the lowers possible because they wouldn't come here. And they are a smaller pool so have more bargaining power so can demand higher wages. So immigrants would be coming to this country on inflated wages which prevents opportunities for our own people to become skilled because they are spending their time doing the work that immigrants used to do so don't have the time to educate and train themselves.
To be honest, I've just chucked all this together in 10 minutes - feel free to pull it to pieces, it may not all be wholly coherent. On the other hand, the referendum was a simple binary choice where you don't have to consider the multitude of consequences because the bus told me all I need to know.