Did winter finally come to Westeros?

Question

In Game of Thrones, several dialogues mention the lengthy winters and the fears they impose on people. We know that the weather system in Westeros is quite different from our world, and each season may last for ages. From the beginning of the first season, the Starks' motto was winter is coming, which made the impression that a long and cold winter will be inevitable. This article says

In the Song of Ice and Fire books, lengthy winters are feared in part for their association with supernatural threats. According to Westerosi legend, the white walkers first appeared 8,000 years before the events of Game of Thrones, during the Long Night—an icy season so severe it shut out daylight and stretched on for a generation. That’s cool. But unpredictable weather is more interesting as a factor for what it can illustrate about the characters we’re already invested in, and how it might affect the melee for the crown.

And I vaguely remember that somewhere in the show someone said there has been a long time since the last winter, so the upcoming winter is likely to be very long and hard (I don't remember which episode exactly). In the final episodes of season 7, we saw that the winter has officially come. But in season 8, especially after the third episode, it seems that winter is no longer mentioned or feared anymore.

So what happened to the long-expected winter? Did it simply pass after the death of the night king?

Addendum

There are some valuable info in this Q&A as a background. And I came across this answer with a very interesting quote:

“Oh, my sweet summer child," Old Nan said quietly, "what do you know of fear? Fear is for the winter, my little lord, when the snows fall a hundred feet deep and the ice wind comes howling out of the north. Fear is for the long night, when the sun hides its face for years at a time, and little children are born and live and die all in darkness while the direwolves grow gaunt and hungry, and the white walkers move through the woods”

George R.R. Martin, A Game of Thrones

Answer 1

Did winter finally come to Westeros?

Yes. Winter "officially" arrived in S06E10, The Winds of Winter.

When the seasons of Westeros change, white ravens are sent from the Citadel to all ends of the Seven Kingdoms.

Summer officially ended in S02E01, The North Remembers, thus making it fall/autumn. And then autumn turned to winter in the finale of season 6. Both transitions are mentioned in the series, as shown in the following videos.

Summer to autumn:

Autumn to winter:

---

So what happened to the long-expected winter? Did it simply pass after the death of the night king?

The "long night" is no more, but a long winter is still possible.

Winters come and go in Westeros regardless of the Night King attempting to invade the lands below the Wall. In S01E03, Lord Snow, Tyrion visits Castle Black and is asked by Maester Aemon how many winters he's experienced, with Tyrion responding, "eight... no, nine."

And then from S02E01, during a small council meeting (same moment as the first linked video) Varys makes the remark,

VARYS: The peasants say a long summer means an even longer winter.

That being said, it still seems extremely possible for a long winter to be upon Westeros at the end of the Game of Thrones series; but, I personally don't believe the death of the Night King will halt the winter seasons all together.

As for the "long night" though, that is associated/characterized by the presence and invasion of the white walkers, first evidenced by Old Nan when telling stories to Bran (also S01E03), and then also by the title of S08E03 where the Battle of Winterfell occurs - The Long Night.

To continue Old Nan's quote provided by the OP:

OLD NAN: Fear is for the long night, when the sun hides for years and children are born and live and die all in darkness. That is the time for fear my little lord, when the white walkers move through the woods. Thousands of years ago, there came a night that lasted a generation. Kings froze to death in their castles same as shepherds in their huts. And women smothered their babies rather than see them starve, and wept and felt the tears freeze on their cheeks.

...

In that darkness the white walkers came for the first time. They swept through cities and kingdoms riding their dead horses hunting with their packs of pale spiders big as hounds.

So yes, I would say that a "long night" isn't possible anymore; however, long winters are still game.

Answer 2

It is only hinted in the show, with the end of last episode showing new plant growing north of the wall. We can only conclude that winter was indeed connected to Nightking. We also saw that when he is created, there was no Lands of Always Winter, so it has to be connected to White Walkers.

Outside of the show, I think I read somewhere that the only thing Martin actually revealed is that winter will turn out to be magical thing, not normal season like we are used to. I suppose that's that, then.