Mary wrote poetry in Latin as well as French but I agree with Jack in that Buchanan wasn't influenced by her in his use of Latin. In fact he wasn't doing anything new by writing in Latin. Latin had been the language of the church and actually the language of law etc. Only being ousted by Scots as the language of Scottish gvt in the 15thC. Buchanan came on the back end of a long line of Latin writers. John Fordun wrote his Scotichronicon in Latin about 200 years before Mary. Walter Bower in the early 15thC wrote in Latin. As did Hector Boece in his "Scotorum Historiae" in the early 16thC. Likewise did John Major (Mair) who is described as the greatest of Scottish writers in Latin by J H Millar in his "Literary History of Scotland" - and Millar goes on to point out that Buchanan studied under Major at university in Paris. If Buchanan was influenced by anything it was by his mentor Major and by the long standing tradition of Latin in Scotland. Millar describes the latter three writers as writing in a more cultured form whilst he dismisses Fordun and Bower's Latin as decidedly unattractive. In fact some of the earliest poetry written in what is now Scotland was written in Latin. ie the poetic works ascribed to Columba! Though yes that was church latin.
I think the idea that Scottish literary figures may have written in Latin to try and imitate the English is missing the point that the Scottish kingdom was a part of a wider European tradition with close links to the continent and that the educated classes were as educated as anywhere else. Despite being much smaller Scotland had four universities to England's two at that time plus Scots studied in France etc.

Because of the latin tradition in Scotland then of course the medieval makars who were writing in Scots had their form of Scots influenced by Latin.