It would almost certainly have been Tuscan, yes. Elizabeths Italian tutor was a Piedmontese man, Giovanni Battista Castiglione, but the language you read in her letters is clearly not medieval Piedmontese, which would have been much, much more distant from contemporary Italian (closer in fact to neighbouring Occitain). Instead he would have learned Tuscan at the court in Turin, as was the standard at most local courts in the North at this time.

Elizabeth's letters are in fact pretty easy to read if you speak contemporary Italian. Tuscan had already emerged as the prestige dialect of the peninsula by the 16th century thanks to important writers like Petrarch, Dante, Machiavelli and Vasari which is why it was 'chosen' as the dialect upon which standardised Italian was based. While its true that this 'decision' had not yet been made, the processes which would propel Tuscan to this position were already well underway by Elizabeth's time.