Mercenary companies of the 30 Years War tended to be drawn from one place-- often Switzerland, often from one particular canton. These units fought with a high degree of professionalism-- Swiss pikemen weren't an ad hoc formation, handling the massive pikes in battle was the product of long training, not a pickup band. In the 16th century, we find both the British and the French appointing captains and engaging them to go out and hire mercenaries, usually German. Similarly in the 18th century, Germans had a steady mercenary business-- hired out as companies by their princes, the notorious Hessians of the American Revolution came with their own commanders, than being recruited by an English commander.
The polyglot multinational companies that I think you're referring to are more typical of the Hundred Years War, I'm thinking in particular of John Hawkwood and the White (or "English") Company. Hawkwood was multilingual -- had an Italian wife-- and we've got letters from some of his lieutenants, also Englishmen, like John Thornbury written in Italian. The Company was organized in contingents, often by national origin, and lead by a corporal-- it's likely that these smaller units were somewhat nationally and linguistically homogeneous, could understand each other and their corporal.
Both Hawkwood's and Thornbury's Italian was idiosyncratic-- one author describes it as "French like". Your suspicion of a "pidgin" is on the right track . . . these men of adventure used language to get by, and get by they did; not so much in a pidgin or creole, but in finding a way to make themselves understood by the folks who needed to understand them.
We may therefore imagine Hawkwood and his fellow Englishmen speaking a French-style Italian, which their hosts did their best to understand.
Hawkwood enjoyed, however, speaking his native language wherever he could: with his countrymen and with those Italians who knew it. The Florentine businessman and ambassador Simone di Ranieri Peruzzi boasted that his facility with English helped him develop a good relationship with Hawkwood and gave him an advantage in negotiations. There remains to this day among the peasants in the countryside near Florence the story that the Italian word bistecca derived directly from Hawkwood's requests in English for a hearty ''beefsteak'' whenever he was in that region.
Note that Hawkwood did think that unit esprit de corps was important- he apparently spend a fair amount of money on the striking white uniforms which gave the White Company their name.
Even earlier, we can consider Roger de Flor, one of the more notorious of the mercenary captains, his troop is usually called the "Catalan Company", and was mostly composed of men of Aragon and Catalonia; they'd likely have understood each other easily enough. de Flor was himself born in Brindisi, he's variously known as Ruggero Flores, Rutger von Blum, or Ruggiero da Fiore; the renderings of his name in Spanish, Italian and German give you a good idea of the languages spoken. We've got a good first hand account of the Catalan company's early battles in Ramon Muntaner's Cronica, valuable as an early soldier's memoir
For a contemporary analogy, consider the Detroit Red Wings some years back, at the time that they had a group of five of the most talented Russian hockey players. One or two of the Russians spoke good English - I think Igor Larionov was the most fluent- but most didn't. There was an assistant coach or translator who spoke Russian, and the Russians typically played as a unit- so they could speak easily with each other. In warfare and hockey, there isn't all that much that you need to communicate-- but what you need to communicate has to be clear . . . so over time men who're fighting together, or on the ice together, figure out a vocabulary for the important things.
Sources:
John Hawkwood : an English mercenary in fourteenth-century Italy
Muntaner and the Catalan Grand Company
The Swiss Mercenary System: Labor Emigration from the Semiperiphery