Gemma (ensemble) - Cum adpropinquaret
I think it's Latin, though it's taken from an old Swedish manuscript. This might help http://en.wikisource.org/wiki/The_Homilies_of_the_Anglo-Saxon_Church/XXVIII

I can't really understand a single word even thou I've studied Latin for 5 years. The manuscript you linked is not written in Latin thou, it's Old English. Sorry I can't help you :(

There is a picture in the Swedish Wikipedia: https://commons.wikimedia.org/wiki/File:Skaramissalet.jpg
The text on the picture reads:
Ante sex dies solemnis pasce quando uenit dominus in ciuitatem Yerusalem occuerunt ei pueri, et in manibus portabant ramos palmarum et clamabant uoce magna dicentes: osanna in excelsis, benedictus qui uenisti in multitudine misericordie tue, osanna in excelsis. (this text is available at https://archive.org/details/missaleromanumme01cath , page 133)
Prima autem die azymorum accesserunt discipuli ad Iesum, dicentes: «Ubi uis paremus tibi comedere Pascha?». At Iesus dixit eis: «Ite in ciuitatem ad quendam, et dicite ei: «Magister dicit: «Tempus meum prope est, apud te facio Pascha cum discipulis meis». Et fecerunt discipuli sic[ut?] preceperat eis Dominus, et parauerunt Pascha. (this is from the Bible with a few minor differences)
Cum adpropinquaret dominus Ierosolimam misit ex discipulis... (The last chant is already available on this site so I’m too lazy to type it: https://lyricstranslate.com/ru/cum-adpropinquaret-dominus-when-lord-appr... — maybe there are some differences, I haven't checked)
Unfortunately, I can’t make out any words in the actual song, so I can’t even check if they sing this or something else, sorry xD It could be useful if someone could compare the music notation on the photo and the actual music they use, I think this would help to match the chant to the words.

That's all that I could do:
(Prima pars:)
Alii expandebant vestimenta sua in via
Alii ramos de arboribus prosternebant
Et qui sequebantur clamabant
Hosanna
Benedictus qui venit in nomine Domini
Benedictum regnum patris nostri David
Hosanna in exelsis
Miserere nobis fili David
(Secunda pars:)
(...) ???
I've listened to the second part many times, but it's very hard to understand.
I think it's Latin, though it's taken from an old Swedish manuscript. This might help http://en.wikisource.org/wiki/The_Homilies_of_the_Anglo-Saxon_Church/XXVIII