It is up to the DM.
While researching this question, I have found nothing definitive, but I have found some things that can help guide a DM toward a ruling.
The gods get their power from you, not the other way around.
In the description of Nyx, the realm of the gods, we see this description of the gods and their power:
Though the gods live in a veritable paradise, they can’t sever themselves from the mortal world. To do so would be to lose the faith of their followers, the source of their magic and a power they will not relinquish.
The power of the gods is contingent upon the faith of their followers. Your faith in your god is the source of the power with which the god dispenses blessings.
The influence of the gods of Theros extends only to Nyx, the Underworld, and the Mortal Realm.
In the introduction to Mythic Odysseys of Theros, we see:
Notably, though, these gods have influence only over Theros and the two planes connected to it: Nyx, the starry realm of the gods, and the Underworld, eventual home of all die.
A particular god's active influence certainly does not extend beyond these three realms.
The DM must decide if the god’s blessings are dependent upon constant connection to their follower, or if once given, they are intrinsic to the follower.
If we rule that the blessings depend upon active communion with the god, then planar travel to any plane other than Nyx or the Underworld will sever that communion.
But if we rule that a blessing becomes an intrinsic or innate power of the blessed creature, that it depends only on the constant faith of the blessed follower, the blessings would continue to function while trekking to exotic planes of existence.
It can go either way, I think.