I think people need to understand that all politicians (of every persuasion) tell lies. It's what they do to get elected. Anybody who casts their vote on the basis of what politicians have told them is an idiot.
In the case of Brexit, both sides told lies, uttered half truths and deceived the electorate. I made my mind up in 1992 that I would vote to leave if ever given the opportunity. Nothing any politician has said between then and the referendum caused me to reconsider and nothing that has happened since the referendum has made me regret my decision. If you believed that the money we previously gave to the EU would somehow just be paid over to the NHS and if you believed the government would stick to its manifesto pledges then you need to get out more. The Remainers forecast all sorts of things would happen should we vote to leave. Few of them have materialised, just as the £350m a week to the NHS hasn't. I've recovered from the lies spouted by both sides very quickly because I believed neither of them. That way I was not disappointed.