Taking Trump first, the truth of the matter is we cannot honestly judge his success because I am not entirely certain his criterion is the same as ours, the world at large. To us success would look like a brighter, safer or richer future for us at least ... and ideally both sides would take away a small trophy. To Trump, success appears to be the amount of media coverage, good or bad, that he manages to generate for himself. In this, if we are honest, he is spectacularly successful. Truth, lies, promises, agreements and breaking them are all tactics for negotiations. How successful such an agreement will be in the long run remains to be seen. That aside, what are the lessons here?
The stronger you appear to be, the more powerful your position is undoubtedly one of the main lessons. In this, one's wealth and military might are definitely among the most important assets one needs on one's side. America against little old North Korea should theoretically be a NO CONTEST. But human spirit and bravado and one or two nukes help Kim Jung-Un's position considerably. Hence for years he managed to hold back the entire world. The support of a powerful friend (China) was definitely a plus.
But in the Brexit negotiations, what has Britain really got that the EU can't get for themselves? NOTHING. Britain cannot win. Not only are they many times the UK's strength and numbers, they have something on their side that UK does not. Their strength, their negotiating position, their demands are the same as their foundations. Originally built to strengthen Europe and avoid conflict in the future, they decided that would build a self-contained economic world primarily to protect itself. To participate in this world you had to give up part of your personal freedoms in favour of a quasi-democratic system. So be it!
Britain seems to think its a country club. Pay your dues and then demand top quality service for your money. This very 'English' quality of being so arrogant you are incapable of understanding your enemy has been at the heart of many a British conflict in the past. From the American War of Independence to India and South Africa, and more recently the Middle East; it wasn't till they realized that militarily they would lose, that they finally decided to give way.
Britain has no real experience in negotiating, and didn't gain any during this Brexit exercise because they are continuously misreading the signs. They are like a new poker player that was advised to wear sunglasses and then promptly goes out and buys mirror lenses. At least if they stayed in the EU they could leave the negotiating to those who are better at it than our crack'ed' team.