There was always worry in Great Britain and the United States during the first two decades of the Twentieth Century about a possible naval war between the United States and Japan, the rising naval power in the East. It, too, was riding on its 1905 victory of Russia, decided largely by that famous naval battle at Tsushima. Such a conflict would pull Great Britain into the war on Japan's side, via the Anglo-Japanese Alliance began in 1902. It was anxiety of this possibility, which neither the US nor Britain wanted, that compelled the UK to allow the treaty to expire in 1921. So to answer you question in full, no, I don't think German buildup drove the British into a closer friendship with the United States, because British concerns about that buildup was its ability to balance the strength of the Home Fleet with the strength of those abroad. Those abroad fleets existed to protect Britain's colonial holdings, something the US was none to fond of, and would not have been particularly interested in protecting on Britain's behalf.