I'm just saying that access to infinite cheap labour has been a boon for large corporations who don't care about the welfare of their employees at all.
You need to look at the whole picture.
Where low paid services are publicly funded (for example home care), wages will not go up because the tax take that funds those wages will not go up.
In the private sector, businesses whose model relies on low paid and underhoused EU labour e.g. Pret etc are much more likely to downscale than to increase wages. There is a limit to how much we will pay for cups of coffee etc.
Automation is the real threat to thousands of these jobs. Look at what's happening in Tesco, for example.
You are forgetting the enormous wage subsidy provided by the state via universal credit et al, and the Claimant Commitment to find 35 hours of work, or else forfeit benefits. Unfortunately, this is only likely to get sharper and more painful for those on low wages after Brexit unless we find a magic money tree somewhere. There is nothing in the universal credit system to incentivise employers to provide better wages or skills enhancement. The benefit system is a key reason why Brexit will not result in higher wages for low paid workers.
I suspect JC is too much for the British electorate, even with a disastrous Brexit. His plan to roll out sectoral collective bargaining is supposed to be the answer to the problems you describe, but it relies on a leap of imagination (not to mention practical barriers) that I suspect is unlikely to take root.
It seems much more likely to me that post Brexit, we will end up with hung parliaments and for the poorest, an entrenched low wage, part-time insecure labour market.