Have you watched Alexander Bett's talk on TED TALKS?
He discusses the post-BREXIT future for the UK. The first thing he says is needed is civic education - not just the crummy courses on citizenship in schools (although schools play a part) but part of everyday life.
The UK could usefully look at what Germany did after WW2 as part of its denazification programme, when it had to undo 12 years of brainwashing. When I first went to Germany in 1970, I was amazed how much people knew about political processes and realised that I knew very little about the processes in my own country. Germany set up a programme which affected every aspect of civic life, as the Nazis had done before them.
I suspect that the British don't care that much, because we haven't had the kind of political upheavals within living memory which mainland Europe has experienced. We've bumbled along. People moan about taxes or prices,etc which might or might not inform their voting choice, but their votes don't count for much anyway, so they don't think much about politics.
I'm quite amazed how little people seem to know about how the EU operates. The information is there for anybody who has ever looked for it.