Great Britian does in a sense have a written constitution, or at least a written document that has helped provide government with a framework. It is the Magna Charter. It is true that there is no document in British history that can be pointed to as a Constitution. What Great Britain does have is a 600 history of "common law." This is the body of court decisions that British courts have handed down on legal disputes. These decisions have interpreted and defined the "law" in both civil and criminal matters. This common law system is also the basis for the U.S. legal system.
In the U.S. our federal and state constitutions have given rise to many legal disputes over the years. State and federal appellate courts have issued written opinions interpreting constitutions as to what they mean in certain areas of the law. As we look at "the law" today what we have is a compilation of court opinions that have "precedent" for resolving similar disputes that come up in the future. This system in practice is very similar to what has developed in Great Britain. In both systems the law today depends in large part on what courts have ruled over time.
In the 18th Century Parliament passed a series of laws that limited the power of the Crown, which had first been limited by the Magna Charter. These "documents" had the effect of limiting the power of Kings and Queens, who are not considered in the modern world to have absolute power. We refer today to these systems loosely as "constitutional monarchies." The common theme between the British system and the American system is the ideal of limits of executive authority. In practice, the systems work in a very similar fashion. This evolution is due to the effects of the legal systems.
In both systems government ultimately works because people accept the authority of government. In the U.S. this is more directly due to the existence of Constitutions. In both countries, however, customs, statutes, and traditions ensure the continued existence of stable government. One of those traditions is the idea of civil rule over the military There are many examples of "constitutional democracies" around the world that have fallen to military action. If democratic ideals are not firmly entrenched written rules mean nothing in a society.
I would write first about the legal history of both countries and then focus on the similarity of the two legal systems. I can help you in the future in any government course.