Although the arthurian legend has its roots in British soil, it was widely known and disseminated on the continent. Since you mentioned Spain in your question, I am going to focus here on how arthurian material was known throughout Spain from the Middle Ages up until the Early Modern period. Going back to the Middle Ages, then, Henry Thomas argues in his classic study Spanish and Portuguese Romances of Chivalry (p. 22-27) that the Matter of Britain was present in the Iberian Peninsula in some form from at least the 13th century. The evidence of this is two-fold: on the one hand, poets such as Dinis of Portugal or Alfonso the Wise were already referencing characters from arthurian romances in their works. On the other hand, baptismal records from this time period also show that arthurian names were fashionable among the nobility: we see girls being named Iseu, Genebra and Viviana; boys, for their part, would be baptised Lançarote or Tristão. Paloma García furthermore, in her article "Arthurian Material in Iberia" (p. 14-15) mentions several other names in medieval Iberian documents of all kinds. We see names such as Artús (which would become our Arturo), Martín Merlín and even Galván.

The question being begged here would perhaps be the following: how did arthurian material make it into Iberia in the first place? Besides the troubadours, translations of French romances - in particular the Vulgate cycle - were the major source of arthurian dissemination in the Peninsula. In the same article mentioned above, Paloma García explains that the earliest translations of the Vulgate cycle in Iberia were done into Catalan in the early 1300s. Of the translations discussed, two are of utmost importance with regard to Spain: the Lançarote, from the French Lancelot en prose, as well as the Tristany de Leonis. These two works go on to provide the backbone of the Spanish chivalric tradition. Due to their influence and popularity, the works that we call libros de caballerías - literally "books of chivalry" in English, similar to what we know as chivalric romances - would go on to be incredibly indebted to the arthurian tradition.^(1)

Now, the most popular and influential of the libros de caballerías was one called Amadís de Gaula (~1492), probably best known to people today because Don Quijote was always minded to imitate its protagonist. Rafael Ramos calls this book an "imitation of the adventures of Lancelot and Tristan" which "succeeded in producing a masterly combination of the chivalric, amorous and fantastic adventures its author found in the French texts," thus "creat[ing] an original story" ("Amadís de Gaula," p. 364). The anonymous author of the first book of the Amadís shows his indebtedness to the Matter of Britain from the very first chapter. Describing the death penalty in Britain, the narrator tells us that "this cruel and terrible custom lasted until the coming of the very virtuous king Arthur ['Artús' in the Spanish original], who was the greatest king to reign there" (trans. & emphasis mine; see p. 243 in Cacho Blecua's edition). There are other references to characters from arthurian romances throughout the Amadís, but this one ought to suffice for our purposes. The popularity of the Amadís cannot be overstated: its incredible influence on the libros de caballerías (and on other literature, such as in Golden Age theater) led to later works in the genre adopting arthurian characters. Just to name a couple of examples: in the romance Florambel de Lucea (1532) bk. 3 ch. 8-10, the Knight of the Vermillion Flower runs into King Arthur and Morgan le Fay, receiving from the latter the sword Excalibur. In the prologue of Don Olivante de Laura (1564), Antonio de Torquemada explains how he saw a procession in which "after [Charlemagne and the Twelve Peers] came King Arthur, accompanied by Sir Tristan and Lancelot of the Lake, with many other principal knights which in the Quest for the Holy Grail are named" (trans. and emphasis mine; see pp. 18-19 in the Biblioteca Castro edition). Arthur is a recurring character in these books, and one finds him where he's least expected.

Finally, it is to be understood that the libros de caballerías were widely read and disseminated. Don Quijote provides us a very picturesque example of this: in chapter 32 of part 1, the innkeeper Juan Palomeque says that "when the time of the harvest comes, the harvesters come together here, and there is always one amongst them who knows how to read; the literate one takes one of these books [i.e., a libro de caballerías] in his hands, and more than thirty of us surround him, listening to him read with so much pleasure that all of our cares fade away" (trans. mine). In his book Romances of Chivalry in the Spanish Golden Age, the hispanist Daniel Eisenberg discusses the reception of these books in a chapter titled "Who Read the Romances of Chivalry?" As shown, their reception - and therefore some knowledge of King Arthur - was vast. Some well-known fans of romances in Early Modern Spain were st. Teresa de Ávila, the humanist Juan de Valdés, st. Ignacio de Loyola and Pedro López de Ayala among others.

Notes:

1 - The term libros de caballerías is hard to translate. They were pseudohistorical, and using the English term "chivalric romance" already keys us in to the fact that they're fiction. However, since they were simply called libros - as were nonfiction books at the time - contemporary readers were not always immediately aware of their nature as fiction.

Bibliography:

The Arthur of the Iberians: The Arthurian Legends in the Spanish and Portuguese Worlds, ed. David Hook. University of Wales Press, 2015. [In this volume are the García & Ramos articles]

Eisenberg, Daniel. Romances of Chivalry in the Spanish Golden Age. Juan de la Cuesta, 1982.

Thomas, Henry. Spanish and Portuguese Romances of Chivalry. Rutledge Reprint Series, 2010.