According to this article The British public still have no idea what they voted for with Brexit – it’s not elitist to admit it, it is claimed:
A survey carried out in the run up to the 2015 general election revealed that 69 per cent of Britons confessed to having “no interest in politics” while a staggering 59 per cent couldn’t even name the then-prime minister, David Cameron.
It links to this article http://blogs.lse.ac.uk/politicsandpolicy/what-are-the-implications-of-political-ignorance-for-democracy/ but as far as I can see it's just from people who do surveys for vouchers. I can't find any other polls of this nature.
