Actually there were two that I found. The first united England and Scotland, the second, England and Ireland. Let's take a look at the circumstances that brought about these unions.
Throughout the 1600's there were plenty of occasions and plans for uniting England and Scotland. Each of these failed. The English point of view was that they had little to gain by uniting with Scotland. The Scots on the other hand did not want to become a “minor” partner to England.
It took a war with France (who else?) that caused the English to have a change of heart. Good ol’ King Louis XIV recognized that a (God forbid) Catholic who went by the name of James VIII, 'the Young Pretender', as the rightful heir to both the English and Scottish thrones. This caused much consternation among the English. . By something called the Act of Settlement it was decided that if Queen Anne should die childless (her only son had died in 1701), then the crown would pass to Sophia of Hanover, a cousin of Charles I. The Scots refused to agree to this and passed a series of acts that were hostile to England. The most important of these was something called the Act of Security in which the Scottish Parliament would have the power to nominate a successor to Anne within 20 days of her death and that the successor did not necessarily have to be from the House of Hanover. This raised the possibility of a separate Scottish monarchy. The English were indeed, not very pleased with the situation. You see, Scottish soldiers were important in the army and the English were engaged in the War of Spanish Succession. Since Queen Anne did not immediately sign the Act of Security, the Scottish Parliament withheld the supply of Scottish soldiers until she backed down.
It’s now 1704 and Scotland is heading more towards independence rather than union. Queen Anne and her cronies in the government were trying to bring the two countries closer together. In order to put pressure on the Scot, they passed something called the Alien Act of 1705. Basically, the Act gave the Scots until Christmas Day to either accept the Hanoverian succession or they were to appoint a commission to negotiate a union. Should they do neither, they would be treated as aliens in England. This meant that any property that was held by Scots in England would be threatened. Any trading arrangements with England and the colonies would de destroyed. Lo and behold, money suddenly became available to Scottish MP's to support union. In a poorly attended Scottish Parliament it was decided to appoint the commissioners to negotiate a union. Talks began ons began in April 1706 and were for the most part, completed by July. When the terms of the union were published, it was passed by a large majority (despite the huge protests of the Scottish people) by the Scottish Parliament on 16th January 1707.
By the terms of the Act, Scotland and England were to become one country. Anne became Queen of Great Britain and the throne passed to the House of Hanover on her death. There was also economic union with free trade on both sides of the border. Scotland was given only 45 seats in the House of Commons and 16 seats in the House of Lords. What can only be described as a bribe of £398,085-10s English pounds was paid by England as compensation for Scotland now taking on part of England's national debt. partly due to English interference. Scotland was to keep its legal system and Kirk.
Alas, on 28th April 1707 the last Scottish Parliament (until 1999) was dissolved by the last Scottish monarch.
The next one brought Ireland and England, shall we say, "together."
In 1800 the Act of Union was passed by both the Irish and British parliaments in the face of much opposition. It was signed by George III in August 1800 to become effective on 1 January 1801. William Pitt intended to follow the Act of Union with other, more far reaching reforms, including emancipating the Catholics, but was rebuffed by King George III, who refused to break his Coronation Oath to uphold the Anglican Church.
Basically, the 1801 Act of Union stated that:
Ireland was to be joined to Great Britain into a single kingdom, and to be called the United Kingdom of Great Britain and Ireland
The Dublin parliament was abolished. Ireland was to be represented at Westminster by 100 MPs, 4 Lords Spiritual and 28 Lords Temporal, all of whom were to be Anglican.
The Anglican Church was to be recognized as the official Church of Ireland.
There was to be free trade between Ireland and Britain.
Ireland was to keep a separate Exchequer and was to be responsible for two-seventeenths of the general expense of the United Kingdom.
Ireland was to keep its own Courts of Justice and civil service.
No Catholics were to be allowed to hold public office.
There was to be no emancipation of the Catholics
The Act sort of backfired. Trying to rule Ireland directly from Westminster solved little or nothing. True, it had it benefits during times of war but it never addressed the grievances in Ireland over land, religion or politics. It also ignored problems with the Irish economy and served as a rallying point for Irish Catholics to voice their objections to British rule.