The first translation of Consolation of Philosophy, the King Alfred version, was written in English around the mid-tenth century. Then another in the 11th-century, a third in the 12th-century copy in Scotland, and a 15th century Latin translation.
"The (King Alfred) translation is a fairly free adaptation of Boethius and some parts are greatly summarised from the original." - https://en.wikipedia.org/wiki/The_Old_English_Boethius#First,_prose_translation
Except the original doesn't exist, nothing exists before the King Alfred version, so what is that referring to?
Was the source of the 15th century Latin translation King Alfred's English version?
If the King Alfred version of Consolation of Philosophy is not the primary source of everything that came after it, then what is?
And if it is the primary source for all that followed, that means everything we know about it is based on that translation.
To answer this question, you need to understand a bit about the original language of the work, and how texts were transmitted and circulated in the Middle Ages.
The "original" being referenced is the original Latin text; Boethius' work, like the works of many Roman authors, was widely copied starting around the late 8th and 9th century in monasteries in the Carolingian world. These scriptoria are widely credited with "saving" many classical texts, whose earliest extant copies date to this era. Later copies were copied from these texts, and then copies made of the copies, and so. The Latin text of the Consolation of Philosophy exists in multiple extant manuscripts from the 9th century, and from studying these manuscripts together scholars can gain an excellent idea of what the 6th century text most likely looked like. (Without a manuscript stemma at hand, I couldn't tell you if all of the earliest extant copies are thought to have descended from a single version of the text or from multiple copies). It's important to know that very few extant copies of works by any classical author actually survive from the classical period. Most survive in later copies.
Here's a lovely 10th century copy of Boethius (in Latin) from Trinity College, Cambridge Note all of the commentary between the lines and in the margins. Commentaries can be another way of determining how individual manuscripts are related to each other. https://www.bl.uk/collection-items/boethius-in-latin
So, the way we know the King Alfred translation was a summary of Boethius' original Latin is that we can compare it to the multiple extant copies of his Latin text that predate the Alfred text and understand how the Latin text was adapted And it should be mentioned that Boethius' Latin text was enormously popular in the Middle Ages, continuing to be copied and commented on in Latin for many centuries. (I've actually held a manuscript containing Boethius' text (among other works) in my hands, although I wasn't interested in it for the Boethius).
Modern scholarly English translations seeking to present Boethius' original work would be based on the Latin text, not on the Old English version.
Incidentally, I'm not sure I understand what the "15th century Latin translation" is," unless it was a retranslation into Latin from the Old English. The link provided doesn't give any details about this particular text.
What do you mean 'no copy survived' and 'Nothing exists before the King Alfred version'?
There are literally hundreds of Medieval manuscripts of Boethius. His Consolations is one of the absolutely most popular and common non-liturgical medieval texts in existence. Not only are there older texts than the Alfred the Great's Translation, there are preserved copies that are older than King Alfred himself, such as MS Vat. Lat. 3363 (early 9th century). I'm not sure what you're asking?