Walpole was not very good at Latin. In fact, in a quasi-famous incident, Walpole attempted to quote Horace. He made a basic error in his Latin and was called out by a political rival. They bet a guinea on who was right. One of the officers of Parliament was called to get the Horace and look up the quote. Walpole lost. The political rival then donated the guinea to a national collection along with a message encouraging pupils to better study their Latin.

It's not clear if George I spoke English prior to arriving in England. There is some evidence he knew at least a little and his mother was undoubtedly fluent. Regardless, if he did not speak English, he definitely learned it while on the throne. We know he also spoke German and French fluently, as well as decent Italian, Latin, and Dutch. When he communicated with most people, and felt his English weak, he would use French. The majority of the nobility also spoke French so George I could communicate with his court quite easily.

But Walpole did not speak French. He does not appear to have spoken any language other than English very well. Thus when we hear of Walpole and George I communicating in Latin, we hear of Walpole speaking bad Latin. We hear that this is necessary because the King did not speak English and Walpole spoke bad Latin and no French. We also hear that he apparently sometimes spoke Latin so his servants would not understand him, with the wry comment that the Latin speakers barely did either. This was a dig at Walpole who was not considered especially educated or highborn. (In fact, Walpole only spent two years at college.)

English government business would have been conducted in English in the 18th century. People would have known a smattering of other languages. We see Pepys (from the period just before Walpole) knew a little Spanish, French, and Latin. He sometimes uses them to conceal his affairs, in case his wife ever read his diary. It's comprehensible but not very good.

The pattern, then as today, was that the more educated and upper class would speak more languages. Since everyone in the upper class spoke a wide array of a limited number of languages, people would try to find a language in common or use a translator. It's very difficult to generalize about this because it's a number of specific stories. For example, we know Queen Catherine de Braganza studied a little English in preparation for her marriage and suffered due to the language barrier for a few years at least. Charles II and the King of Denmark, however, were able to speak to each other even while uproariously drunk.