I agree there must be a lingua franca
or maybe more than one. What it/they might derive from is anyone's guess. Didn't the Orions used to have a widespread empire long, long ago? Maybe lots of folks speak Orion. There was a reference in "The Trouble With Tribbles" about folks learning to speak the Klingon tongue, while members of the Klingon crew in STiii clearly spoke in something other than Klingon to each other...sometimes.
Evidently, English immigrants have taken over France. At a convention Marina Sirtis expressed great pleasure at this realization.