I haven't seen any indication that EU migrants cut wages, the data indicates it isn't true, see the lack of correlation between migration and changes in UK citizen wages in the fourth chart on this page:

http://www.ft.com/cms/s/2/0260242c-370b-11e6-9a05-82a9b15a8e...

Also, Britain is not America. Real wages in the UK consistently go up across all income levels, and have done over the last 30 years, partly because of 'left-wing virtue signals' like minimum wages, and working tax credits.