When Nymeria came to Dorne, it wasn't exactly an empty land.
First Men were the first to settle Dorne and after them, came the Andal invaders with their seven gods.
Nymeria and Rhoynar were the last to migrate to Westeros. When she came, Faith of the Seven was already established in Dorne. Most of the Rhoynar refugees were women and young men who intermarried with the Dornish when their Princess did.
It is said that, amongst the Rhoynar who came to Dorne with Nymeria,
eight of every ten were women … but a quarter of those were
warriors, in the Rhoynish tradition, and even those who did not fight
had been hardened during their travels and travails. As well,
thousands who had been boys when fleeing the Rhoyne had grown into
manhood and taken up the spear during their years of wandering.
By joining with the newcomers, the Martells increased the size of
their host by tenfold. When Mors Martell took Nymeria to wife,
hundreds of his knights, squires, and lords bannermen also wed
Rhoynish women, and many of those who were already wed took them for
Thus were the two peoples united by blood. These unions enriched and
strengthened House Martell and its Dornish allies.
TWOIAF: Ancient History - 10,000 Ships
So it was by marriages that the Rhoynar in Dorne converted to the Faith of the Seven over the course of the coming years. That however does not mean all Rhoynar gave up their religion, the Orphans of the Greenblood still worship the Rhoyne.
"It is a long journey," Arianne said, "but it will go easier once we
reach the Greenblood. Some of Garin's people will meet us there, the
orphans of the river. They live on boats, and pole them up and down
the Greenblood and its vassals, fishing and picking fruit and doing
whatever work needs doing."
"Aye," Garin called out cheerfully, "and we sing and play and dance on
water, and know much and more of healing. My mother is the best
midwife in Westeros, and my father can cure warts."
"How can you be orphans if you have mothers and fathers?" the girl
"They are the Rhoynar," Arianne explained, "and their Mother was
the river Rhoyne."
Myrcella did not understand. "I thought you were the Rhoynar. You
Dornishmen, I mean."
"We are in part, Your Grace. Nymeria's blood is in me, along
with that of Mors Martell, the Dornish lord she married. On the day
they wed, Nymeria fired her ships, so her people would understand that
there could be no going back. Most were glad to see those flames, for
their voyagings had been long and terrible before they came to Dorne,
and many and more had been lost to storm, disease, and slavery.
There were a few who mourned, however. They did not love this dry red land or its seven-faced god, so they clung to their old ways,
hammered boats together from the hulks of the burned ships, and became
the orphans of the Greenblood. The Mother in their songs is not our
Mother, but Mother Rhoyne, whose waters nourished them from the dawn
AFFC: The Queenmaker
George R.R. Martin also attributes this religious change to intermarriages:
Linda: Do (or did) the Dornishmen follow gods other than those of the First Men or the Andals, because of the Rhoynar? If so, does this
have anything at all to do with the ruling Martell of Dorne calling
him or herself Prince or Princess?
George_RR_Martin: The Dornishmen may have a few odd gods lying about, but they were pretty much assimiliated by intermarriage, so
the Seven dominate down there too. The "prince" was the style of the
ruler of their original cities, along the Rhoyne River.
Web Chat with Fans - Citadel link here
Unlike Dorne however, North was never conquered by the Andals and the dominion of the first men continued there until the day Aegon the Conqueror decided to unite Westeros. There was no major change in demographics even before or after that and that's why the Old Gods still hold sway in the North unlike Dorne where the Rhoynar were mostly absorbed into the locals.