The yes" side says "of course they do, more labor supply means lower wages." The no side says, "If a million people migrate to the UK, who is going to sell them food and clothes, build them houses, teach their children? When the population is larger there is more demand for workers."*
Luckily, we have economists to untangle the mystery for us. And over at The Financial Times, Tim Harford does exactly that. "It is fair to say that economists do not really agree," he says.
Well, glad we cleared that up.
*I guess I should say that my read of the evidence is that both things are true: Immigrants increase wages in the aggregate while exerting a mild downward drag for unskilled workers.