I've been researching, and I've found that American colonists argued against taxation without representation because it violated the UK Constitution. However, when searching for the UK Constitution in 1776, I've not found anything. I know there was one though, what was the UK constitutional rule about taxation and representation in 1776?
The Declaratory Act, 1766, of the Parliament of Great Britain, stated the constitutional position regarding taxation in the colonies. It said Parliament "had, has, and ought to have," full power and authority to make laws to bind the colonies in all cases whatsoever. The phrase "all cases whatsoever" included the right to levy taxation. Most acts of parliament, or other legislatures, make changes to the law, effective from a certain date. A declaratory act is different in that it does not purport to change the law, but merely to state clearly what the law already is. In this respect, a declaratory act is somewhat akin to a court ruling.
The Declaratory Act of 1719, relating to Ireland, asserted the right of the Parliament of Great Britain to make laws binding on Ireland. However this did not include the phrase "in all cases whatsoever", and was never understood or interpreted as allowing taxation to be imposed on Ireland. All Irish taxation had to be approved by the Irish Parliament. (This is up until 1801 when Ireland and Great Britain were united into a single kingdom with a single parliament.)
There was no mechanism for any act of parliament in Great Britain to be challenged as unconstitutional. From a legal standpoint, as seen from Great Britain, it appears Parliament therefore did have the constitutional authority to tax the American colonies in 1776,
However it could be argued that Parliament was wrong in regard to the Declaratory Act, as respecting the existing position, and powerless to alter the position going forward. Over the centuries there were several occasions when English monarchs were alleged to have behaved unconstitutionally and exceeded their powers. Parliament asserted this on several occasions and insisted that parliamentary approval must be obtained for laws and taxes. The constitutional crisis of the 1760s and 1770s differed in that the suggestion now was that parliament, not just the king, were exceeding their powers in relation to the colonies. Parliament was now "villain" rather than "victim".
The dependency of taxation on representation was a corollary of a deeper rule. Fundamentally, the principle was that taxation required the consent of the people on whom it was levied. The Magna Carta (originally 1215 ) was held to forbid taxation except by common consent of the realm. In later centuries it came to be accepted that the way in which common consent was given, or withheld, was through the summoning of representatives to parliament. So, for example, the Bill of Rights of 1689 actually refers to parliamentary approval rather than common consent.
Prior to the 1760s internal taxation within the colonies was purely a matter to be determined by the local assemblies (e.g. in Virginia the House of Burgesses). Whether the king or Parliament had, or had not, always had the right to tax the American colonies, in the century and a half since they were founded they never actually had taxed them. Custom is an important part of the British constitution. King George III in his Coronation Oath had sworn "to govern the people of Great Britain, and the dominions thereto belonging, ... according to the respective laws and customs of the same". The fact that never before had a British government taxed the American colonies was a strong constitutional argument that the colonies enjoyed freedom from such taxation as a customary privilege, and therefore a constitutional right. (There were parallels with Ireland, where taxation undoubtedly required Irish Parliamentary approval.)
That taxation required consent, and that consent required representation, was generally accepted. This can be seen from the concept of Virtual Representation, argued by George Grenville and others. George Grenville was Prime Minister from 1763 to 1765, when the Stamp Act was introduced. He did not question the constitutional imperative for representation. Rather, he argued that the Parliament of Great Britain, not just as individuals but as a body, represented all British subjects worldwide. True, the American colonists could not vote, or stand, in elections for Parliament; but neither, at the time, could most Englishmen. William Pitt, on the other hand, ridiculed the notion of Virtual Representation, pointing out that the MPs, lords and bishops were familiar with the condition and needs of the people in the districts they represented, and usually known personally by them; but that none of them knew anything about the situation in America.
Both sides accepted that representation was constitutionally imperative to consent to taxation, differing only in the form it should take.
So yes, taxation without some form of representation was unconstitutional, but it was arguable whether virtual representation sufficed. The Declaratory Act was clear that parliament did have the right to tax the colonies. However, the departure from custom in levying tax on the colonies was a failure to govern in accordance with American customs, which was arguably unconstitutional.
The nebulous nature of an unwritten constitution, based partly on law and partly on custom, means that no certain answer can be given. This was one of the reasons for the United States eventually opting for a detailed written constitution, to minimise uncertainty, and a supreme court to deal with any that remained. This way constitutional crises could be avoided. The American Civil War, almost a century later, in which each side argued that constitutionally states did, or did not, have the right to secede, demonstrates it was only a partial success at avoiding dispute.
In the UK disagreement over the extent of the royal prerogative (now exercised by the government), between government and parliament on the question of revenue, persists (see FT and Guardian ) leading to cancellation of an increased probate fee which had been planned for May 2017.
The UK has never had a single-document constitution like the US one. There are occasionally moves to create one, but they never get anywhere.
There are a number of acts of parliament of fundamental constitutional importance, such as the Bill of Rights (1689) and the Act of Settlement (1701). There are also some key court decisions, which are parts of the common law, and the Royal Prerogative (nowadays exercised, in practice, by the government). There are conventions that have evolved over time about the operation of Parliament, and several books that record and generalise constitutional principles that are regarded as reasonably authoritative. But there is no document that you can refer to that tells you what the whole constitution is.
This has some surprising advantages. It is far easier to create a constitutional crisis than is the case under a formal constitution, which means that everyone is careful not to do it and a degree of consensus is maintained, which limits the scope for abuse of power. The constitution can also change gradually, by consensus, and changes that don't work can be reversed easily. And being literal-minded about the constitution, or anything else, just isn't respectable.