Before the shock and magnitude of the leave vote had really sunk in, Theresa May was jetting off to the US to hold hands with Donald Trump to beg him for a trade deal. He agreed, apparently with great enthusiasm. “We could have a really, really good trade deal,” he confirmed by Twitter. Great news to Brexiteers still euphoric at the result they had just pulled off. The problem is that Trump didn’t say who this deal would be good for, although the clue is in his campaign slogan: America First. Further information can be found in his book The Art of the Deal, wherein he explains that you always make a deal with your opposite number when they are vulnerable because this allows you to win by making them lose.
Whatever one thinks of our various international trade regimes, the underlying philosophy is that if we can prevent a large group of countries from having trade wars then everyone benefits. Obviously it's important how we define "everyone" but the idea is that every country, in some sense, benefits. True or not, that's the idea. But if you take yourself out of the club, or one of the clubs - as the UK is doing - it doesn't follow that other countries, or the club you just left, can't benefit by totally screwing you, or at least by using their greater bargaining power to strike a deal that benefits them (or at least politically connected industries) more.
The UK will have almost no bargaining power. All trade agreements have a thousand different industry groups and their lobbyists screeching about the specific quotas and tariffs that their little niche industry requires. That's part of the justification for trying to Fast Track trade agreements (the other is to keep them secret as long as possible to sheild politicians from accountability...), because otherwise such agreements would be impossible. Again, that doesn't mean that system is Good, it just means that if the UK removes themselves, in part, from that system, by removing themselves from the EU, they're going to have to deal with negotiating with every country. Good luck!