Almost half of LUMW is dedicated to the aftermath of the battle against Chaos. A large part of that concerns the consequences of the Gods not properly disposing of their enemies' corpses, which lead to the birth of the Skaven:
But here was their great mistake. For the bodies of the Chaos hordes still lay on the battlefield, a vast carpet of carnage. And on top of it all lay the hideous bodies of the Chaos Gods themselves. As they lay there rotting, there came what always must come to battlefields: a plague of rats to feast upon the dead.
[...]
And so it was, when Ulric returned to the battlefield, he saw the rats feasting, and realised his great mistake. In their great consumption the rats had taken on a remnant of those foul gods' power, and become like them: a new race, like Men and the Dwarfs, only made from pure Chaos.
In response to this Ulric summons a great flame to burn what bodies remain:
Although it was too late, Ulric struck his hammer hard on the stony ground and brought forth a great flame with it. And with that flame he burnt all that foul offal that remained to ashes.
and also He asks Manann to drown the battlefield, but I want to focus on the flame. After the bodies are disposed of, Ulric is described as becoming the patron God of humanity, teaching them how to fight and how to work metals and giving them the gift of fire:
He taught them how to mould steel to make warhammers and swords and axes, and taught them how to wield them. He taught them how to fight, and to hunt, and to kill. And lastly he taught them how to make fire and how to use it.
The very final paragraph of LUMW brings up Ulric's fire two more times.
In return, the people of Ulric pledged to never suffer a Chaos-thing to live while they had breath in their bodies, and to ensure that every beast of Chaos that fell would be burned to ash, cleansed from this world by Ulric's Holy Fire.
[...]
We strike without fear against the ratmen, and the Beastmen, and all the creatures of Chaos, and raise the cleansing flame of Ulric to all the heavens above.
Now, this entire closing section is full of pretty obvious bullshit. More reliable sources date the birth of the Skaven race to the fall of Tylos, an event that happened thousands of years after the Coming of Chaos. Historical records tell us that the knowledge of steel comes from the dwarves and not from Ulric. Perhaps most egregious is the fact that Chaos Gods are claimed to be among the corpses - I'm pretty sure the Four are still around. If you want to be super charitable you could try and read all of this as somehow metaphoric, but I don't want to be charitable: it's pretty clear that the myth is attempting to paint Ulric as responsible for all the Good Things and as the protector from all the Bad Things.
Having established that everything else in that part of the myth being deeply suspect it's now time to talk about the flame of Ulric.
Why does Ulric have a scared flame? It doesn't exactly fit a God of winter, and yet it's brought up quite a few times - but only at the end of the myth. Could this be another mistake/lie? Well, there's a certain Flaming God that's been cut out of LUMW but which does appear in the other myths:
MF: Just as it seemed all was lost, a great, white fire erupted from the Throne, and Asuryan the Phoenix, wearing a bifurcated mask of white and black, strode forth.
OS: Then, just as the Four and their allies arrived for the Final Battle, Flaming Phoenix, whom all had thought dead, returned from atop His Gleaming Pyramid, and He smote about Him.
It's not hard to guess what's going on here: in the original myths Asuryan shows up at the very end of the battle against Chaos to smite the opposition with flames, but in LUMW Asuryan has been cut out of the myth so that very important role is instead given to Ulric. Ulric's sacred flame is merely an appropriation of the Flame of Asuryan, the cleansing flame of the Emperor of the Heavens.
There's just one tiny problem with this theory: the Sacred Flame of Ulric exists. It's an actual thing in Middenheim, you can go and look at it if you want. So what gives?
The similarity between Ulric's flame and Asuryan's flame goes beyond their role in the myths; as Cadaeth pointed out in her conversation with Mathilde, both are ever-burning flames that burn only the unworthy. Indeed, the two most famous instances of the Flame of Ulric not burning someone are Magnus and Sigmar, who would both go on to be worthy Emperors, which is quite reminiscent of the Flame of Asuryan not burning those who are worthy of being Phoenix Kings.
It could be that the Flame of Ulric has nothing to do with the Coming of Chaos at all, and LUMW is merely an attempt to make sense of the flame - legend has it that Ulric lit the flame by striking a mountain for some reason, so LUMW gives a reason and makes it a pretty good one. On the other hand OS also places Ulric at the Coming of Chaos, and the flame does seem quite similar to Asuryan's. I think there must be some connection between the two flames, but what could it be?
As one possible explanation, note that flames can be lit from the Sacred Flame of Ulric and carried elsewhere (there's one such flame in Ulrikadrin). What if the Sacred Flame of Ulric was itself lit from another flame - specifically, Asuryan's flame? Ulric and Asuryan fight on the same battlefield, then after the dust settles Ulric goes up to the fires left from Asuryan's smiting of the Chaos hordes or whatever and somehow makes it His own. The flames are similar because they are pretty much the same flame, except that Ulric somehow gave His flames a bit of His nature or something. Not a very satisfying explanation imo, and it doesn't really explain why Ulric was the only on the battlefield with the idea to get a cool flame.
I have another possible explanation. Asuryan is said to live in a great pyramid, topped by a diamond throne on which He sits. The actual Shrine of Asuryan, where the eternal Flame of Asuryan burns, is a pyramid. MF also mentions what appears to be Asuryan's pyramid, seemingly as the site of the battle against Chaos:
Verena eventually agreed to join the survivors at the Great Pyramid.
[...]
Step-by-step the defenders were driven up the pyramid, until there were only a handful of Gods about the Diamond Throne at its top.
Maybe Asuryan's pyramid is an actual physical thing? An artefact of the Old Ones, perhaps one which has the power to summon flames which burn the taint of Chaos, which is why the defenders rallied to it during the invasion. The MF myth is largely about Hoeth attempting to use an Old One artefact against Chaos, and maybe Asuryan likewise took that artefact or He might have been the one in charge of it or whatever, and perhaps there were other similar artefacts - perhaps one for each of the five cities, or one for each of the two polar gates - and one of the others belongs to Ulric.