I would assume that most of the major powers have an "official" language which a significant share of the population throughout their empire can speak reasonably fluently. The further you get from the regional capitals and centers of trade and power, however, the less of that you'll typically see. After 300+ years, most of the core worlds of each empire should have converged toward only moderately different dialects of the official language, and in some cases those could be almost indistinguishable. When you've got mass entertainment in another language, it does affect local speech over time through the incorporation of stray words and phrases, and any planet with a solid HPG connection is going to receive enough communication and mass-viewing content to draw the languages together over time.
Movies and TV programs created for nationwide consumption have unified the language and dialects within the US to a surprising degree just over my lifetime. There's still a noticeable difference, but it's much more subtle now. I can still make fun of them over it, and expect that they'll do the same, no real offense intended or taken.
Note that English gradually drifted away from its largely Germanic origins as it became heavily influenced by more local Gaelic and French cultures as well as Latin, but the drift of both England's English and American English have been fairly similar over time, largely because of the strong cultural interaction between them, and both sharing some of the same outside influences. Basically, pre-Shakespearean "Olde English" is just about as foreign to an Englishman as to an American. Most residents of the two modern cultures can converse normally with those of the other country, aside from a few stray words, or occasional awkward secondary shades of local meaning to an otherwise identical word (example: beyond its application to animal carcasses and pillows, "stuffed" in America also refers to having over-eaten, while in England it takes on the secondary meaning of "pregnant"; aside from a smooth surface, "flat" in America refers to a punctured tire, where in England it's an apartment). Most of those instances would be easily understood through context or a brief moment of confusion and an explanation, and most content intended for interplanetary consumption would very likely provide that as a matter of course after the first awkward "incident" or two.
I would expect that the "official" languages spoken within MOST of the major factions of the Inner Sphere (not necessarily as their primary language) would eventually end up about as close as America's, England's, and Australia's versions of English, with the more remote locations drifting to something half-way in between that and whatever the strongest local influence might be. Some, like the FWL, would likely retain much more internal variation than those of House Kurita.
