Oh boy, my wife is writing her PhD dissertation on exactly this topic (well, trans-Atlantic correspondence networks of exiled loyalists to be specific) and so I have been learning a lot about it as a result as well. To sum it up, thousands of loyalists ended up leaving during and immediately following the war. Many of them spread throughout the British empire, with many parts of modern day Canada actually being founded or heavily populated by them. Places like Prince Edward Island, Nova Scotia, New Brunswick and Ontario ended up with large loyalist populations. Others settled in the Caribbean, and some smaller numbers ended up in Great Britain itself.

Now, a lot of these loyalists ended up suffering economically, as their properties and businesses in the former colonies were seized. Many petitioned the British government for recompense, saying they deserved to be compensated for remaining loyal to the crown. Unfortunately for them the British government did not quite see it that way, and while they did provide some aid, most loyalists ended up poorer as a result of the war.

Further, many loyalists found themselves not welcome in other parts of the Empire. They clung to the idea of an British identity and expected that they would be free to settle in other colonies or in Britain and resume (as much as possible) living their lives and conducting business as usual. The problem is that those people who already lived in those colonies often saw these new comers as competition. This led to frosty receptions in some places.

Finally, those these exiles remained loyal to the Crown, most had been born in North America, and as such saw places like Boston or New York or the Carolinas as "home", even if they were now a different country. As such, after a decade or so, you see a lot of loyalists actually returning to the US. One topic that keeps coming up over and over in the letters my wife studies is exiled loyalists writing to contacts and loyalists who have already returned to the US about what sort of risks they might face if they were to return to the US. Would they be arrested by the American government? Were the people still hostile to them? Would they be able to get their property and businesses back? The answers are pretty varied.

If you want to know more, I'm sure I can get my wife to answer in more depth later this evening.

Sources: Liberty's Exiles by Maya Jasanoff, Tories: Fighting for the King in America's First Civil War by Thomas Allen, lots of archived letters from/to/between loyalists, and talking to wife as she rants about her research.

EDIT: My wife has said she'll come on and answer your questions and correct whatever I may have gotten wrong tonight. We'll be out with friends for a bit, but should be back by 10pm eastern or so.