British colliers sustained the Great White Fleet

How did the Great White Fleet obtain enough coal without extensive British cooperation?

It did not: most contracted colliers were British-registered because the American merchant marine could not meet the fleet's immense supply requirement.

Federal rules favored American-flag supply ships, but the United States lacked enough merchant carriers willing and able to move the required coal. American operators also struggled to find return cargo that would cover the cost of the voyage. The Navy therefore contracted thirty-eight ships to supply roughly 125,000 tons of coal between Hampton Roads and San Francisco. Only eight were American-registered, while most of the remainder sailed under British registry. That dependence looked awkward for a display of independent American power, yet it also showed Japan and other observers that Britain was prepared to support the United States. Battle-fleet mobility depended on merchant shipping capacity and access to friendly carriers.