In addition the comments by @Brendan about the selection process and legal distinction between the United States Constitution and the British parliamentary structure, it seems that this question attempts to get at the reasons for, and potential consequences of, the following question:
- Given the importance of the Separation of Powers doctrine between branches of government throughout the world, what maintains a legislative check on the executive in the United Kingdom, given that he is selected from the legislative body?
- In practice, how do the legislative and executive branches operate efficiently given the overlap in membership?
First, the reason for this construction is fairly easy to uncover given the evolution of government in the United Kingdom throughout history. Unlike the United States which had a hard and fast split from the United Kingdom and a deliberate period of establishing an entirely new government, the government of the United Kingdom has grown and evolved for centuries into what it is today. This gradual evolution from a tribal society, to a pure monarchy to the democracy we see today is the main reason the United Kingdom does not have a distinctly independent executive. Ever since the signing of the Magna Carta in 1215 the people of the United Kingdom have been trying to wrestle more power away from the monarchy and into the hands of the people. This led to the creation of Parliament with the power to pass some laws. Eventually, that power extended to all laws, including constraints on the monarchy. However, as the monarchy existed throughout this period, and continues today, even as the executive power transitioned away from the monarchy, the task of selecting a new executive came from the one established power source that existed, Parliament. This is the same reason that unlike many modern democracies, the United Kingdom does not have one constitutional document, but rather a collection of laws that together make up the basis for their government. The United States on the other hand had the opportunity to start with a clean slate, with the opportunity to lay out all of their institutions in the best way the Founding Fathers could think of, and they wanted an independent executive in order to help differentiate themselves from the system they had just rebelled against.
Determining the potential consequences of the executive being so intertwined with the legislative branch is a bit more nuanced. Both nations (the United States and United Kingdom) believe that the power of the executive should be held in check by a strong legislative branch and allow the legislature to pass laws limiting the power of the executive and even remove them from office (impeachment proceedings in the United States and a vote of no confidence in the United Kingdom). The real difference between the two systems is that the United States has a two party system and the United Kingdom a multiparty system. In the United States, where there are only two dominate parties and the majority has enough votes to pass legislation on their own the power of a legislative executive could be almost unlimited. However, in the United Kingdom where the majority party may not have an outright majority and coalitions between multiple parties both in opposition or in forming a government are often required, the power of the majority party is much less. This makes the power of the opposition to force a vote of no confidence a much more plausible event than impeachment proceedings in the United States and leads to a more practical check on the executive's power. For example, in the United Kingdom today the government is formed by a coalition of Conservatives and Liberal Democrats with a Prime Minister and Deputy Prime Minister from different parties.
In addition to the head of the government, the real work of the government in both countries is handled by cabinet secretaries and ministers and both countries have restrictions on these offices. In the United States, a person selected to serve in these offices must immediately resign any other elected office they hold. Similar rules exist in the United Kingdom as well as limits on how many can be selected from the House of Commons, House of Lords, etc. This ensures that the work of both branches of government can continue effectively and allows for a real distinction between the branches of government, despite the overlap at the Prime Minister level.