Anglo-Norman was never the main administrative language of England: Latin was the major language of record in legal and other official documents for most of the medieval period. However, from the late 12th century to the early 15th century, Anglo-Norman French and Anglo-French were much used in law reports, charters, ordinances, official correspondence, and trade at all levels; they were the language of the King, his court and the upper class.During the 12th century, development of the administrative and judicial institutions took place. Because the king and the lawyers at the time normally used French, it also became the language of these institutions. From the 12th century until the 15th century, the courts used three languages: Latin for writing, French as the main oral language during trials, and English in less formal exchanges between the judge, the lawyer, the complainant or the witnesses. The judge gave his sentence orally in Norman, which was then written in Latin. Only in the lowest level of the manorial courts were trials entirely in English.
During the 15th century, English became the main spoken language, but Latin and French continued to be exclusively used in official legal documents until the beginning of the 18th century
https://en.wikipedia.org/wiki/Anglo-Norman_language
As far as I know, most early US colonies exclusively used English, not French, at a time when British courts operated in French. So it seems not unreasonable that British English has more French influence than American English because French was the official language of the British courts for centuries and it never was here.
A notable exception in the US is that Louisiana has French influences of various sorts, including on its legal system. But most of the US is very much British influence and English language based.