According to the film's official novelisation, the USCSS Covenant has the ability to communicate with Earth at FTL speeds, via a series of relay beacons that it's been dropping off at periodic intervals.
A quick check of the displays inside his helmet indicated that all the relevant systems were operating normally once more. That meant that though it would still take the usual interminable period of time to do so, they could once again make contact via the established relay system that led, like a string of electronic beads, all the way back to a now very distant Earth.
While the Covenant was completely self-contained, as was necessary for any colony ship, that thin thread of contact with home remained important as a link to the planet they called home. As they journeyed onward and the ship continued to automatically drop off a relay unit to extend the system at each recharge stop, it also meant that once they were established on Origae-6, those back on Earth would be able to learn that the colony had successfully established itself at the chosen destination.
Contacting the Earth from the location of their accident will take longer than two weeks.
“She’s close.” As curious as any of them, Ricks had already run the necessary calculations. “Real close. Just a short jump from our present recharge position. Maybe a couple of weeks. At jumpspeed, it wouldn’t even be necessary to go into hypersleep.”
...
She could only file her formal objection, knowing that it would take more time to reach Earth than it would for the Covenant to arrive in the uncharted system—much less for a response to come back to the ship. Given that reality, she wasn’t sure it was worth the effort to file.
