So what would have happened if from 1770 to today if Britain had given it’s American colonies seats in Parliament?
Unlikely, more likely a dominion or commonwealth or even a seperate kingdom would be created.
We’d all be speaking Canadian.
They would all have been imprisoned for misuse of apostrophes. This is the main reason that people emigrated, not tax. That was just a cover story.
Then Britain would have held on to North America – but the Americans would be in Parliament, deeply involved, all the time, in British national and imperial business as well as American business. That would be fun to watch. I wonder what side they would have taken on the Corn Laws? Of course, with a growing population, the Americans would always be for Parliamentary reform. In fact, once the Dominion of America’s population surpasses that of the British Isles, we have an American Empire! 
Also, this would have set a precedent for future British colonies. At least the British/white people in them would have expected representation in Westminster at an early stage . . . and then, eventually, the natives would start demanding the vote . . . and the Commons fills up with brown faces.
“So, when you say ‘American Indian’…”
And it is kind of fun to imagine Franklin, Jefferson, Adams, Hamilton, etc., as MPs in 18th-Century London! 
Well, that’s another thing . . . Once the political trend became clear, it might have discouraged the British from colonizing places that couldn’t be made into “settler states” – for fear all these non-British peoples, once colonized, and once given votes and representation, would overwhelm the culture of the Empire. E.g., if the Empire includes India, then, very eventually but inevitably, we will have what amounts to an Indian Empire that happens to include Britain and America and have its capital in London.
Of course, if the British Empire decided the go the even-more-imperial route anyway and worry about political problems later, then it could always get help from the Dominion of America. Or, at least, it could always get troops, as Royal American Army units or as recruits for the Imperial British Army or whatever. And sailors for the Royal (or Imperial) Navy. All that wicked and glorious British military adventuring and colonizing in Asia and Africa in the 19th Century, the “British Century,” would include Americans, and their relatives back home would feel included. At the same time, I have no doubt America would still have plenty of resources and energy for its own push westward (unless the Imperial Parliament – with more American representatives in it every decade – decreed limits). So the “manifest destiny” element of our culture would still be there, but we would think of it as bound up with the larger British destiny.
Considering how proud and smug and patriotic-chauvinistic and expansionistic both the British and American national cultures were in the 19th Century in our timeline, that would make for some fearsome synergy.
It would have required a major political change. Britain and America had substantially different demographics. There was, for example, no aristocracy in America - so who would be representing America in the House of Lords? Would Parliament attempt to create an American noble class in order to stock the House of Lords? Or would Americans accept be shut out of the upper half of Parliament?
Even in the House of Commons, you’d have problems. The American ruling class (those represented in the colonial legislatures) were essentially businessmen and professionals. So if America were sending MP’s to London, it would be these same people. But English businessmen and professionals were under-represented in Parliament. English MP’s were from the upper class. So you’d almost certainly end up with a political crisis when British businessmen saw their American counterparts holding political offices that they were effectively excluded from.
Another factor to consider: If the American Revolution is averted, then the French Revolution (precipitated by a state financial crisis precipitated by, among other things, the cost of French aid to the American rebels) is also averted or, at least, delayed. The underlying problems would have remained . . . but the revolution might have happened a bit differently, and might have turned out entirely differently, if it had started a few decades later, in the early railroad age. (Steampunk novel!) Possibly, nothing like the Napoleonic Wars happens.
Actually, a title would be a good thing with which to bri-ahem secure the lasting loyalty and friendship, and honour the contributions to society in America and in the Empire, of leading Americans. The British political leaders of the day were really good at that sort of thing. The Puritans and Quakers might not have proven quite so resistant to such Vanities as you think. 
Yes.
Yes, you would. 
So the pressure for Parliamentary reform is more and sooner. Egged on at all times by American MPs, many of whom, culturally/ideologically speaking, are all still for this democracy thing, anyway, especially Lord Jefferson. Maybe, with American support in Parliament, the Chartists get what they want when they want it. Now there’s an AH worth thinking about!
I think that the most likely result would be that, around 1776, the Americans would get fed up with the taxes and regulations that were being imposed on them, and would fight and win a war for independence. Of course, they probably would have done that if they hadn’t had representation, too, but there’s no way to know for sure.
Oh, wait, you mean if we had representation in Parliament that was elected by the Americans ourselves? Then you’ve got to posit a few more layers of hypotheticals to get all the Brits represented by representatives they elected themselves, first. Who knows where that would have led. Unless you’re assuming for some reason Parliament would give the far-away Americans privileges not afforded even to the locals?
Another consequence is that the slavery issue (at least, within the English-speaking world) gets thrashed out in Parliament – by MPs from slave provinces of America (including not only the cotton provinces of the southeastern mainland, but the Caribbean sugar islands of Jamaica and Barbados), and MPs from free provinces of America (including Newfoundland, Upper Canada, Lower Canada, etc.), and MPs from Britain and Ireland (where only Protestants can vote, and one wonders – one really wonders – what the American MPs from various provinces might have to say about that!). Without the precedent of the American Revolution to inspire them, it is kind of hard to imagine the slave provinces of America trying to secede from the Empire over the slavery issue. Especially knowing they’d have to fight Royal American and Imperial British forces. So, Civil War averted.
I would assume that Parliament, just to save trouble, would allot a certain number of seats to each colony and tell the colonial legislature to send that many to London, let them worry about how to pick 'em. At first, anyway. Eventually there might be pressure in America for more direct elections.
And I do think that would be enough to avert the Revolution. The colonies’ leaders now would have vested interests in the Empire and personal political careers bound up with it. (It’s one of the things that holds the U.S. together, every governor wants to be president someday.)
I think a delay in the French Revolution is inevitable; the masses in France won’t have the example of the American Republic, but they will remain starving, freezing, and suffering just the same, and will certainly revolt eventually. The question in my mind is whether Napoleon will be able to rise to power in a Royal French army as easily and quickly as he did in the Republican French army-- his family were minor nobility, which might work either for or against him, but he was a Corsican, and it’s hard to picture Corsicans doing quite so well in a Royal army. I think most likely Napoleon remains an artillery officer in the French army. What he does when the French Revolution starts, being older and presumably more entrenched in this counterfactual than in history, is difficult to predict. You’re right; it’s quite possible the Napoleonic Wars might never have happened at all.
Plus, a lot of the resentment and tension in the colonies was between the appointed governor and the elected legislatures (if they were allowed to vote on anything). The reform would have to go a lot further than a few token seats in parliament. I think the taxes floating down from on high were just the straw that broke the camel’s back.
The actual issue of taxes was really pretty minor. This was a couple of hundred years before Grover Norquist was born. Keep in mind that even at their highest point, the taxes Parliament imposed on Americans were substanitally lower than the taxes that Britons paid.
The main problem was that Americans saw themselves being ruled by what increasingly appeared to be a foreign power. No American would ever sit in Parliament. Worse yet, no American would ever have a real role even in American government. Men like Washington and Adams and Franklin saw themselves being ruled by English-born officials who were appointed by London. They resented the fact that they were politically powerless in their own country. They weren’t so much mad at the taxes as they were mad about England’s ability to declare Americans would pay taxes without any American having a say in the matter.
OK, I think that takes care of the late 1700s, how about we move into what would have happened from 1800-1899.
Okay, let’s say that the American colonies were given proportional representation. And let’s say that the parliamentary reforms of 1832 were adopted a few decades earlier.
You’d end up with a majority of MP’s from Britain with a substantial minority from North America (along with Scotland, Ireland, and maybe some other parts of the Empire like the West Indies). The American MP’s would probably be seeking westward expansion into the American continent. The English MP’s would see this as an unnecessary provocation with other European powers. So there would be an issue there. Assuming there was a war with France (a safe assumption because France and England were always having some war or other) the Americans would be far less concerned about the safety of the home islands and maintaining a fleet in being in the channel. They’d want the Royal Navy to be out in the Atlantic protecting trade and maybe seizing French colonies. On the plus side, a war with France would give the Americans an excuse to seize Louisiana.
The next big crisis would be over slavery. Parliament formally ended slavery in England in the 1770’s, which was easy because it was virtually non-existent anyway. The bigger struggle was over ending slavery throughout the British Empire. With a strong slave-based economy in the American south, there would have been much strong resistance to abolishing slavery. But overall, the majority of the Empire was opposed to slavery and they would have almost certainly banned in in the 1830’s or 1840’s (historically it was 1833). The southern provinces might have tried seceding but they would have been in an even worse position than the southern states were, fighting the rest of America plus the British Empire.