Carryfast:
Firstly we know that the immigrants are coming here because incomes/living standards are better,or perceived as being better,than in their own countries.In which case increasing wages without closing the door will just create an even bigger attraction and corresponding increase in the levels of immigration.
But you fail to account for how finding employment is instrumental in achieving that better standard of living. The homeless and jobless in Britain are not better off than the employed in Eastern Europe. Migrants will only come here en masse if the job offers are there for them - and those offers won’t be there if employers aren’t able to undercut, because without undercutting, employers will already have an adequate workforce to meet their needs.
We are more or less reaching that point anyway at the moment (albeit for different reasons), where wages and conditions have deteriorated so far in the industry that there is a “shortage” of workers willing to undercut any further, and so employers are having to make do again with the pool of workers they have and actually retain those workers, rather than abusing existing workers all the way to the exit door knowing that another migrant worker is on his way in to replace the loss and asking 10% less in wages too.
On that note they aren’t under cutting the minimum wage because they can’t.
But drivers aren’t generally earning NMW. Employers are undercutting the going rate which is higher than NMW, and they are also undercutting in the ratio of effort to pay - in other words, they can’t pay NMW workers any less per hour, but they can whip them twice as hard for each hour’s pay, and throw them onto insecure contracts, and so on.
What they are doing is adding to the labour supply thereby holding the minimum wage/wages in general to an artificially low level.While your idea would obviously result in us being swamped by more immigrants.In which case the answer is close the door thereby creating an economic environment in which wages will rise by by both market forces and making a higher minimum wage sustainable.
Which leaves the question what is your actual agenda.In calling for open door immigration together with increasing the economic attraction for immigrants even more.Which can obviously only result in more immigrants trying to come here. 
But the point is not to make British employers less attractive to migrants. It is to make migrants less attractive to British employers. A new migrant with poor English, no local familiarity, foreign training and unverifiable driving experience, is only attractive to an employer at less than the cost of a settled worker for the same work. If the employer has to pay the new migrant the same as the settled worker for the same effort, then why on Earth would he prefer the migrant with all those disadvantages?
Or are you seriously suggesting that migrants have generally better skills than settled workers? By skills I mean something that makes the migrant more productive for the same effort or able to do more things (like fix the engine as well as driving the wagon) - I don’t mean the migrant’s willingness to thrash himself and render extraordinary effort, or comply with every demand like a lickspittle, for what seems to him like a king’s ransom (but is only an ordinary wage for a settled worker).
While ‘if’ you’re actually saying use a massively higher minimum wage to reduce the motivation for employers to import immigrant labour.Then what’s the problem of doing that together with closing the door.
Somehow I’m guessing your intention is the former of those. 
Because “closing the door” doesn’t address any of my concerns, except by indirectly addressing that which I would address directly with employment regulations. And my concern is only with undercutting wages and conditions - I’m not concerned about ‘sending them back’ or ‘keeping them out’ beyond that, because I like migrants all the same as I like settled workers, and I think a reasonable dose of them is a good thing for our society.
