The problem is that our labour market has opened up to include people who are used to living on a very much lower wage than here, mainly because they come from a place where the cost of living is much lower, and are happy to endure worse living standards here for the money. Industry has exploited that.
I suspect the UK has benefited in that Sainsbury's now make more profit on a pint of milk than they did. They are still screwing the dairy farmers into the ground on the purchasing side, so the benefit hasn't gone down the supply chain. Up to the shareholders and CEO, then. Great.

We could increase the minimum wage, but that only exacerbates the problem of cost of living and widens the gap between us and the overseas labour markets. In any case, employers find other ways of getting round the minimum wage, especially for migrant labour where the employees don't know their rights and they can work under the radar.
The same story is happening in all domains around the EU. It was a group of smaller markets, all different, often quite radically different, but coexisting with enough resistance to mobility that they coexisted happily. Remove the resistance overnight and we have all sorts of problems, like trying to run the economies of Greece, Portugal, Spain, Italy and Ireland on an economic model that works just fine for Germany.
The EU has put everything into the same pot without allowing enough time (possibly a generation or two are required) for things to level out naturally. In some cases the differences are more cultural, so probably won't work themselves out even given time. Can you ever see Greece or Spain matching German levels of productivity? Can they do so without being culturally ruined in the process? Is there a need for them to do so, other than to make a fundamentally flawed currency union project work?