Err... OK, gonna be that guy...
Bronze and Iron weapons are not that different in hardness nor the sharpness of the edge they can hold. In point of fact, Iron is arguably inferior to Bronze as a material for weapons since it rusts and tends to shatter rather than simply bend when damaged or used against something harder than it (a bent Bronze weapon can easily be repaired while a shattered Iron weapon needs to be completely reforged). As far as usability Iron is strictly worse than Bronze too, the big importance for Iron is that learning to work with it is the gateway to Steel which IS superior to Bronze for weapon development but in 1000 BC I think the only Steel in existence is the rare "Meteoric Iron" (as an aside, this is where the trope of "magical meteorite iron" comes from, as basically in this era if you had a weapon that was made of Meteoric Iron you basically had a Steel sword before the concept of Steel was even a thing and Steel's properties were such it seemed magical.
Basically if Bronze is entirely ineffective Iron should be as well. Or both are only marginally useful.
So, other things to consider: how much food do these creature need to consume. If it's a considerable amount, they're going to have trouble crossing the Eurasian Steppe (though, I do recognize that's where you'd drawing these critters from mythologically), as you have them originating in China (where all the worst plagues upon the world tend to originate from*) that implies they have a fairly high nutritional intake requirement. So the question is how fast do they spread? Further the question of if the knowledge of black powder developed in China can get across the Silk Road BEFORE these creatures start to show up in the Western regions.
The immediate changes are significant though. Given their being endemic to China it is likely China ends up VERY different culturally and politically. One of the chief responsibilities of government there will be in hunting and killing these creatures and it will hinder the growth of Chinese philosophy as there will simply be less people and prosperity to allow for thinkers to actually be a thing. The long term effects of this means that East Asia would be fundamentally different down to its roots than the modern world and likely unrecognizable (this means that Japan too, while not having to deal with these creatures, is ALSO unrecognizable considering how much culture, philosophy, and religion they drew from China).
Another major factor involved in their spread is going to be their resistance to cold (you seem to imply that they are not highly resistant to it as you mention them spreading south and west from China, but not North into Siberia which has a lot of biomass for them to eat even if lower population dense of humans).
Folks here saying that the Chinese will flee to Japan and survive there. No they won't. There is no China, not as anything recognizable. In 1000 BC the
Zhou dynasty had barely been formed, and what we recognize as cultural and philosophical China would not exist for around another 500 years. Further the Zhou were not a seafaring nation, barely controlling any of the coast of China and were mainly located in north-central China. As such they won't be successfully fleeing to Japan, which will remain under the control of the Jomon. Heck IRL the first Chinese influences settlers to Japan, which were from Korea, did not show up until 900 BC anyway.
The further information concerning the lack of ability to burrow and climb actually means mountains are going to be a considerable impediment to their spread. What this truly means is that humans will retreat into the mountains and turn them into fortifications, leaving valleys and plains to the worms. This means that there will be many holdout city-states throughout mountainous regions, and it's potential that the entire Tibetan plateau might go generations before they get invaded. They will also have trouble entering the Middle East, due to the geography of Asia (and as they originate in China, there's a good chance they won't be adapted well to the heat and lack of water in the deserts, thus further slowing their progress into Africa). However, Northern Europe is generally screwed. There's no major obstacles between western China and northern Europe, as they're basically the two ends of the Great Eurasian Steppe, which is basically the natural environment for these creatures. Southern Europe, especially Italy and Greece will actually be somewhat protected from them due to the mountains slowing down their spread there, as well as Iberia (though France is screwed). Ironically, I have my doubts about these creatures being able to infest Africa, as there's no good migration route from Asia into Africa for them. You have the mountains slowing them into the Middle East combined with the desert there and the Med forming a major barrier to entry from Europe. Even if they manage to get into North Africa, the Sahara Desert is basically an impassible barrier for them and Ethiopia is again on a major plateau that will prove difficult for them to infest, thus making the eastern route south difficult.
I imagine they eventually would get into Central Africa and likely overrun things there, but it will take a while.
Meanwhile humans will be basically driven into the mountains for protection from these creatures, and cities will generally always be walled or built on high areas surrounded by cliffs. This is actually advantageous for humans to retreat to as these mountains often have mineral wealth needed to eventually build the weapons needed to kill these things. You'll likely see the development of the long, horse mounted lance earlier here, as even if it's only tipped in Bronze the sheer force imparted by a lance driven by a horse and rider, or potentially a team of horses and chariot, is likely more than sufficient to pierce the hide of these creatures (and if it isn't, no early firearms would be able to pierce its hide either, Lances hit harder than MODERN firearms, to say nothing of pre-modern ones**). Wermhunters will be a major heroic profession amoung the population, likely sponsored by cities and rulers to keep regions near cities free from major infestations.
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* No, seriously, historically as far as we can tell Plague originated in China, and there's a good chance other major pandemics did too, most recently being Covid-19, but that's just the most recent in a long list...
** No, I'm not kidding. F=MA and while the A is lower here you're dealing with a much more massive M. The Force behind a lance is roughly equal to the mass of the horse+rider+kit so you're looking at an easy 1,000 kg (around 2.2 tons) moving at, say, 4.5 m/s (around 10 MPH which is slow for a charging horse) and ending up with 4500 Joules of Force... for comparison the
30-.06 round at most has ~4100 Joules of Force. And the force is concentrated into a similarly small (arguably the Lance might put that force into a SMALLER area than most bullets).
