Someone else can probably provide a more in depth answer to this question, as my interests lie mostly in the American Civil War and British naval history of the late 18th and early 19th centuries, but I can provide a partial answer to get you started.

First as a little background, recall that the United States generally followed a policy of isolationism since its inception. That is not to say that America did not have its imperialist moments (they certainly did: Mexico, Philippines, and Cuba to name a few), but these "takings" were never against British holdings and were generally places that the British Empire had no active interest.

There was no ignoring the growth of the American economy from the mid 19th century onward, but American growth was not necessarily a direct threat to British strength - in fact, mutual growth was probable and beneficial.

Also recall the similarities between the two "empires," they are both English speaking democracies that essentially shared the same philosophical roots.

There are numerous theories around British opinion and policy concerning the United States in the era you are asking about. No one is probably most correct, and a blend of the different theories is the most likely cause of the peaceful transition of power from the Old World to the New.

One to consider is the "balance of threats" concern. The British Empire was facing a renewed threat from the Continent in the form of German unification and nationalism, in addition to increased competition with Continental powers in a second bout of empire building in Africa and Indo-China. The United States was not a large threat to renewed British imperialism, and did not consist of any threat to the homeland - unlike changes quickly occurring on the Continent.

Along these lines comes the second theory, the "status quo" possibility. The United States respected the British led international system, and were not direct contenders or threats to the holdings of the Empire. However, the U.S. did assert what they perceived to be their rights in the New World, and the British Empire did release the majority of their claims in the area as a result.

In sum, the United States did not pose a true threat to the British Empire as it stood at the time, especially as the U.S. was very hesitant to directly oppose the world's largest power. U.S. policy makers were content to expand in areas that did not directly conflict with their English speaking brethren. The British Empire stood to gain economically from U.S. growth, and they could potentially rely on their support in the conflicts brewing on the Continent. Even without assurances of American support, the British Empire could not stand and protest a growing American power while simultaneously keeping their influence in the changing European landscape.

Edit: removed a redundant "at the time"