The various allied Celtic nations in the book (Eirelan, Scotland, Kernow, Wales, and Bretan) share a lingua franca that they refer to as “Gaelic,” though it is not the Gaelic we know today or from former times. It is a language that grew out of the Irish, Scottish, and Manx forms of Gaelic (Goidelic languages), combined with English. The everyday language of Eirelan and Scotlan is quite close in structure and vocabulary to the shared language and thus the Eirelanders and Scots need make little adjustment in speech (mainly avoidance of slang and colloquial expressions) to converse freely with other Celtic peoples. Kernow’s Brythonic-root language has, over time, grown closer to the Gaelic spoken in Eirelan and Scotlan due to extensive military and commercial contacts. The Welsh speak among themselves a much-changed Brythonic language that cannot be understood without study by the Gaelic speakers. Though the Cornish share this language with the Welsh, they tend to be somewhat more fluent in the lingua franca version of Gaelic. Finally, the Bretans (on the coast of France) speak a form of Brythonic related to Cornish but much modified by French and English. The Welsh and the Bretans must learn the shared Gaelic tongue as a foreign language. Many Welsh do; not so many in Bretan whose connections with the island nations are somewhat looser.
The English language survives in written form and is known, studied and read by educated citizens. It is spoken (in a modified form) only in the areas of eastern England outside of Wales, Kernow and Scotlan.
Those interested in deepening their knowledge of these Celtic languages might start with Wikipedia, which has some excellent articles.