It depends what you mean by "court", as the royal law courts and the royal court were more or less the same thing until the preeminence of the Court of Chancery came about in the 1500s. The majority of my knowledge is the legal side of things so I'll talk about that.

Sir Edmund Coke in the preface to First Part of the Institutes of the Law of England (1628) said that Law French, which was the language developed in the court in the 11th-14th centuries, was "commonly written and read, and very rarely spoken", and there seems to be general agreement that the last legal proceedings conducted in French took place well before 1700 and possibly before 1600, even if writing them down was done in French or Latin (there's a famous case from Dyer's Reports from 1688 reporting on a 1631 case in butchered Franco-English-Latin).

The start of the downfall of French in the courts was the Statute of Pleading 1362 which said:

Because the laws, customs and statutes of this Realm be...pleaded, shewed and judged in the French tongue, which is much unknown in the said realm...The King...hath ordained...that all Pleas shall be pleaded in any Courts whatsoever...shall be pleaded, shewed, defended, answered, debated and judged in the English tongue.

This led to the development of Chancery Standard, the language of the government bureaucracy in the aforementioned Court of Chancery, influenced by the knowledge of French and Latin by the people who wrote it, which is the first modern dialect of written English. The Parliamentary rolls were done mostly in English from 1420 onwards, as that article states.

So the time at which English first penetrated the courts was probably some time in the late 1300s and this process was solidified over the 15th, 16th and finally 17th centuries, with it largely being a relic in the law courts alone by 1600, and most of the "French" spoken in the courts was of little resemblence to either the French of the 1200s or modern French, being largely "debased" with English and Latin words (see Maitland below).

Maitland writing in the late 1800s stated that any Law French influence had ended by the time of the 'Revolution', i.e. the Glorious Revolution of 1688. It's telling that the most recent Law French manual until the 1990s had been published in 1779.

However, some French law terms are still visible in English law, namely words such as appellant, respondent, chattel, chose, defendant, estoppel, mortgage, tort.

Interestingly Parliament still uses a minimal version of Law French combined with Chancery Standard proclamations to pass laws upon prorogation - the clerk reads out the Act title then says "La Reine le veult" (middle French for 'the Queen wills it') - from 6:00 onwards.

So to answer your question: officially in the late 1300s; in reality (as a spoken language) some time in the 1400s/1500s; meaningfully by 1600; completely (as a written language) by 1700.