In addition to the source I listed above, take a look at the epilogue to Section 5 (pp. 486-490) in Murray Rothbard’s book, A History of Money and Banking in the United States.
In short, foreign dollar redemption for gold under Bretton-Woods was, in practice, done mainly by foreign central banks. As the U.S. inflated dollars in the 1960s, the dollar devalued against gold and foreign central banks began to redeem dollars for gold. Gold was flowing out of the U.S. Treasury in the late 1960 and early 1970s. This is one reason, Nixon closed the gold window in August 1971.
U.S. commercial bank notes were not redeemable for gold or silver under the Federal Reserve System, which started operations in 1914. Commercial banks exchanged their gold reserves for Federal Reserve Notes and the Fed came to own the “country’s” gold stock. If Americans wanted to buy foreign goods, they just traded Federal Reserve Notes for foreign currencies. They did not need to convert to gold first.
Finally, under Bretton-Woods, foreign governments stored their gold in the U.S. So when a trade imbalance required gold to move from the U.S. to a foreign country, it would simply be moved from one part of the vault in Fort Knox to another.