I might be wrong, but if I'm reading it rightly the proportion of immigrants has to increase by 10 percentage points, it doesn't matter what the proportion of immigrants is overall, its the increase that causes the reduction in wages. I think Missmoon is right when she says that the effect may be siginficant, but the size is small.
The best example of anywhere seeing this level of increase would be Boston, which saw its over all population increase by 15% between 2001 and 2015, with an increase in the amount of immigrants from 1.3 so an almost 10 percentage point increase, so its likely that people in Boston on the lowest wages have seen slightly lower wages over the last 12 years. However, the 10 percentage point increase didn't happen within a year or so, so the effect would have been spread out.
You are correct in noting that other working conditions are not noted, but I'd also factor in the fact that for the lowest paid the tax threshold has risen considerable, and would most likely more than offset any decreases in pay caused by immigration.
Boston is actually a really interesting case study, if you look at their council briefing on the impacts of immigration.
www.boston.gov.uk/CHttpHandler.ashx?id=8079&p=0
Its interesting that the LSE findings that immigration has little effect on native unemployment is true here, as Boston has an extremely low level of unemployment.
However, there are very few other areas in the UK that have had such a large percentage point increase in migrant workers, and it is evident in many of the places that voted leave that there is not this level of immigration. So I find it difficult to see a trend between unemployment and low wage growth and "people voted on what they have seen and experienced themselves". It simply is not the case that there is a big enough immigrant population in places like Hartlepool, Sunderland, Carlisle, Anglesey etc to have had the effects that are being blamed on them.
This then links in with the LSE data that shows that the effect on real wages has been the fall out of the crash of 2008, and that they have fallen across all of the earning percentiles
blogs.lse.ac.uk/politicsandpolicy/real-wages-and-living-standards-the-latest-uk-evidence/
Excellent articles linked by Wrong, some of which I don't agree with, but show that the issues that have effected many people are extremely complex. Which does go along with the idea that the simple answers given by the leave campaign, although effective in gaining votes, are actually not the answers to an extremely complicated problem.