Nothing is going to happen either way for at least two years, assuming Article 50 is triggered immediately (which forces negotiations with the EU). The drops we're seeing in global markets right now is reactionary. It's caused by the same people who throw themselves off of buildings after a bad day of stocks. It will bounce back in days if not within one.
What concerns the Remain camp most is the uncertainty of the negotiations. Despite also being the largest deficit, the EU still imports the most from the UK. Leave voters decided the uncertainty/sending this very clear message to the government outweighed staying in a failing relationship that was only getting worse. The assumption is that the negotiations will fundamentally change the UK's trade agreements to try and balance some of that growing deficit. Whether or not that ultimately succeeds depends on the negotiators.
Consider that we, the USA, are not in the EU, and do not have a trade agreement with them in place (TTIP is the proposal). Our 2016 trade deficit with the EU so far is ~$50B. Our trade deficit with the UK individually is barely $1B. All things considered, I'm not sure the UK should be most concerned here. Take into account recent mass immigration and labor protests, you start to understand why the majority was willing to risk it.