The Brexit referendum had among the highest turnouts of all British votes in history, there was a considerable amount of media produced to explain each side (and many more remain arguments and endorsements were aired on radio, television, and through social media advertising campaigns), and the opinion was nonetheless trending further toward Brexit on the eve of the vote. It is just about as democratic as it could possibly be.
I honestly don't know where this theme of regret is coming from, because among the people I know who supported Brexit, none of them have expressed anything other than ongoing support, and a desire to see the original intent of the referendum respected by the parliament (as it clearly wasn't respected by the Prime Minister).