In the first Halo game, the Elites can be heard speaking their native tongue for a mission or two, then we start hearing them speaking English, and they continue to speak English for all of the other Halo games. My impression was that Cortana figured out how to make Master Chief's suit translate Sangheili during the first Halo game, and that translation program simply continued working for the rest of the series (even when Cortana herself wasn't around).
But during the first mission of Halo 5, where we play as Spartan Locke, Jul 'Mdama's first few lines of dialog are in Sangheili, and one of the other Osiris team members has to translate it for us, which indicates that they don't have any such translator programs in their suits. Then in the cutscene at the end of the mission, Jul is speaking English without anyone interpreting. Later on, the missions on Sangheilios also have several Elites, including the Arbiter, speaking English pretty much the entire time. As far as I know, Osiris team doesn't have any advanced AIs in their heads that could whip up a translation program on the fly like Cortana did.
Assuming this wasn't a mistake, the simplest explanation seems to be that a lot of Elites have learned English by now, and they switch languages based on who they're talking to like any real-world polyglot would do. Since I can't recall any explicit references to Covenant aliens learning English, do we know for sure if this is what's going on?