Minimum wages are threatened by increasingly high executive pay at the top of the industrial hierarchy, which hasn't been accompanied by any increase in productivity. That's why workers are paid poorly. Foreign workers are a tangential issue by comparison-though I'm sure conservatives around the world would love you to blame foreigners and not the overpaid executives at the top.
Even if we had zero immigration, workers pay would still be threatened by automation. I don't see Syrians or Poles replacing British people working in banks and supermarkets, I see rows of machines.