That's a very interesting question. Depending on your definition of "part of" and "United Kingdom", you could pick from several dates: 1200, 1267, 1284, 1535-1542, 1746, 1801, 1998 or 2006.

First, realize that the territory that is known as Wales today was only once brought under a single crown prior to the conquest by Edward I (Gruffydd ap Llywelyn briefly united it from around 1055 until his death in 1063). Otherwise, there was no independent Wales, only a collection of territories under various kings and princes, who spend quite a lot of time fighting each other, when they weren't fighting Normans or raiding Norman territories.

By 1200, the Normans had invaded Wales and established themselves in the south under the Marcher Lords (the area being known as the Welsh Marches). In that year, by treaty, Llywelyn the Great had sworn fealty to King John of England, and around that time Gwenwynwyn of Powys swore allegiance to Llywelyn. So most of Wales, in one way or another, answered to the King of England. In 1267, the Treaty of Montgomery established Llywelyn's grandson, Llywelyn ap Gruffudd, as Prince of Wales, with control over all of non-Norman Wales. Llywelyn swore fealty to King Edward I. So, during this period, parts of Wales were under direct Norman control, and parts were ruled by Welshmen who were vassals of the King of England.

In 1282, the Welsh revolted and Edward I decided to end Welsh independence once and for all. The Statute of Rhuddlan brought Wales firmly under Norman control. It established English criminal law, but Welsh law still prevailed in civil matters. The English crown directly controlled parts of Gwynedd, and the few remaining Welsh princes became Marcher Lords.

The Laws in Wales Acts of 1535-1542 ended the feudal lordships and annexed Wales as parts of England. This made English the language of official business in Wales, and established representation in Parliament. At this point, Wales is not any sort of independent entity. To make sure that was clear, the Wales and Berwick Act of 1746 stated that any legal mention of "England" also included Wales and Berwick-upon-Tweed.

"The United Kingdom of Great Britain and Ireland" was created in 1801 from the Kingdom of Great Britain and the Kingdom of Ireland. This might be the technically correct answer to your question, since this is the formation of the United Kingdom, with Wales legally a part of England.

Welsh Language Acts of 1967 and 1993 gave the Welsh language legal standing in Wales, but it was not until the Government of Wales act of 1998 that Wales had it's own national assembly, and the Government of Wales Act of 2006 finally put Wales on more equal footing with Scotland and England.