Given they were drawn from across Europe, what language would groups like the Knights Templar have spoken?

The language spoke in monastic orders like the Templars of Hospitallers depended on the order itself and the time period.

The Crusader states were originally led by a French elite and colonised by Latins. Due to the predominance of French knights, French was used as the lingua franca (I believe this is where the term gets its meaning, actually) by the upper classes for everyday conversation. At least this was so during the first two centuries of the Crusader states. Administratively, French was used as much as Latin.

I do not have any specific information on the Knights Templar, but I can speak for the Knights of St. John (aka Hospitallers, Knights of Rhodes/Malta). As they shifted their bases of power to Cyprus and Rhodes, it is possible that whilst French was still widely spoken amongst the Knights, Italian could also have been adopted as a lingua franca due to the Ventian influence there.

As time went on, Latin had eclipsed French for administrative purposes and was used in higher-register writings. During the 14th and 15th centuries, the French language was going through drastic changes:- changing from Old French to Middle French. Therefore, a stable language was needed and Latin, whose teachings remained unchanged for 1500 years, fitted this perfectly.

As the Knights' power base moved to Malta (geographically close to Italy and Sicily), Italian was increasingly used as a lingua franca both in speech and writing. I have already spoken about the linguistic changes French was experiencing, and therefore the use of Italian was rather practical. As Brincat (2001) states:

Moreover the codification of the Tuscan variety made Italian an easier language to learn, especially if one had acquired the rudiments of Latin. Of all the Romance dialects, Tuscan had remained closest to the speech of the Classical age because the area between the Tiber and the Northern Apennines had remained protected from linguistic interference, unlike Rome itself which had been sacked many times, and witnessed drastic population movements which had southernised” its speech. Moreover, during the early 16th century Rome itself was undergoing a process of linguistic Tuscanisation under the Medici popes.

Brincat also states that the adoption of Italian as a lingua franca could also have served a diplomatic purpose, as there were intense rivalries between Spanish and French knights.

It also must be said that the Order was organised into different 'langues' or Chapter Houses depending on where Knights came from. The Knights in these langues would almost definitely have spoken their own languages. If you would excuse the wikipedia link, you can find more information on Langues here.

Sources: Brincat, Joseph M. "THE LANGUAGES OF THE KNIGHTS: LEGISLATION, ADMINISTRATION AND DIPLOMACY IN A MULTILINGUAL STATE (14TH–16TH CENTURIES)." Language and Diplomacy (2001): 139-148.