Hello!

This is in my opinion a great question, because it is also a window into the lives of diplomats and diplomacy in general during this period. Latin as a first language died out sometime around the evolution of Romance languages (French, Spanish, Italian, etc.), but it still existed in a form for church usage, for diplomacy, education, science, etc., exactly as you said.

Latin underwent three main evolutions in the 2nd millennium A.D., from Medieval Latin, to Renaissance Latin, and then to Neo-Latin or New Latin, following the Renaissance period. Latin as used in that period was not a first language of any speakers, but it was indeed utilised in a lot of specific contexts. Specifically to answer your question, yes, it was used as a bridge language between Europeans with no other common language, and the situation you stipulated - an educated Pole speaking with an educated Englishman or German in Latin - would indeed have happened.

A good example of the aforementioned is King George I, previously Elector of Hanover, when he became the King of Great Britain in 1714. George spoke no English, but was fluent in French and German. The Prime Minister (the first P.M.) of the United Kingdom, Sir Robert Walpole, Earl of Orford, spoke neither French nor German. However, they could both speak Latin (albeit according to some, badly), and used it to communicate with each other and to govern the nation. I quote William Coxe's 1800 publication, Memoirs of the Life and Administration of Sir Robert Walpole, Earl of Orford, "Before I conclude the reign of George the First, one remarkable fact must not be omitted: As the king could not readily speak English, nor Sir Robert Walpole French, the minister was obliged to deliver his sentiments in Latin; and as neither could converse in that language with readiness and propriety, Walpole was frequently heard to say, that during the reign of the first George, he governed the kingdom by means of bad Latin." This is followed up from the Letters of Horace Walpole, "It was perhaps still more remarkable, and an instance unparalleled, that Sir Robert governed George the First in Latin, the King not speaking English, and his minister no German, nor even French. It was much talked of that Sir Robert, detecting one of the Hanoverian ministers in some trick or falsehood before the King's face, had the firmness to say to the German "Mentiris impudissime!"

So, as can be seen, Latin could indeed be used with some utility, but the length and extent to which it could be used, would no doubt vary from speaker to speaker. As French (and later English) became the main international languages of diplomacy (and were actually spoken as first languages by many people) in the 17th, 18th, and 19th centuries, it became more convenient to use one of those - they are spoken in colloquial, everyday terms, unlike Latin.

I hope my answer could be of some use to you!