Not to be too enthusiastic but the British loved the Great White Fleet. Americas entry into the world of Naval superiority was the best thing before sliced bread to the British who were stretched thin. Consolidating power against Germany at home meant leaving the Pacific and South America with only token forces. The British especially those in Australia, New Zealand and Hong Kong viewed the Great White Fleet as an Anglo sphere force that would keep the Japanese in check.

At this time Japan had just defeated Russia in the Russo Japanese war. Germany was vowing to build a fleet that would rival the British right in Britain's backyard. In fact Germany's strategy said that they didn't have to defeat the British Navy. They only had to damage the British Navy enough that Britain would lose superiority allowing nations like Japan to take advantage of a much weaker British Navy. The US fleet represented a second power that would keep nations from taking advantage of Britain in the event of war.

US British relations had been becoming significantly warmer since about 1894-95 in what is called the Great Rapprochement. There are many factor to this warming of relations not the least of which was the desire to build an alliance between Britain and America against the rising European power of Germany. (France was also trying to warm relations with the US. The Statue of Liberty wasn't just a gift. It was also a subtle reminder that the US and France were free countries. Not like those awful Germans.)

The British greeting in Auckland, Sidney and Melbourne was so enthusiastic that several sailors from the US fleet deserted in each city.

The British definitely reacted to a threat to naval superiority. They did so by building a fleet to defeat Germany and by building alliances with the US and France. This strategy would pay off in WW1 as the US provided supplies and munitions to the British and then when the US entry into the War forced Germany to surrender.

As a note on Dreadnoughts. The US finished plans in 1904 and began constructing their own Dreadnought style battleship the Michigan in 1906. The British in an arms race with Germany completed the Dreadnought much sooner than the Michigan which wasn't completed till 1910. The Dreadnought was the first of its kind but other nations were right behind the British.