The prevailing theory was "virtual representation," where even though people would not be directly represented through elected officials their interests would still be represented by the members of Parliament for all of Britain... and by extension America. In fact, most people in Britain could or did not vote for members of Parliament, and many countered the Patriot's argument with that fact.

Here's a quote from Soame Jenyns, counter to whiggish arguments of taxation without representation: " Lastly, that no Englishman is, or can be taxed, without the consent of the majority of those, who are elected by himself, and others of his fellow-subjects, to represent them. This is certainly as false as the other two; for every Englishman is taxed, and not one in twenty represented: copyholders, leaseholders, and all men possessed of personal property only, chuse no representatives; Manchester, Birmingham, and many more of our richest and most flourishing trading towns send no members to parliament, consequently cannot consent by their representatives, because they chuse none to represent them; yet are they not Englishmen? or are they not taxed? "

Here he is saying that many in England did vote for their representative, yet still paid taxes without a problem. They still had the rights of Englishmen that the colonists were fond of. there are others that say the same argument as Jenyns.

You have to remember the colonies were used to having their own colonial governments where many had the power to vote. Poorer sorts from Britain could emigrate and have opportunity to own land, a qualification for suffrage.

So in short, the British did think the colonists were represented, the colonists just didn't think it was the way they wanted it.