In English, "thread" can be both a mass noun and a count noun. It's fairly obvious that "thread~threads" (count noun) corresponds to uzi~nyuzi, but how would you translate the mass noun sense, as in "I need (some unspecified amount of) thread"?
Indeed, yarns or threads would indicate to me not quantity, but different kinds of thread or yarn.
"Yarns", definitely, but I can talk about a single thread or multiple threads without it meaning "a kind of thread", as in "my shirt has a thread loose", or "tie the two threads together" (by comparison, to get the same meaning I'd have to say "tie the two pieces of yarn together").
That, of course, is absolutely true. It is always so hard to think about usage without any context.
You can tie loose threads together, but yarn is a different story (no pun intended). Yarn is singular, but in the plural we use balls of yarn, skeins of yarn, pieces of yarn, or kinds or colors of yarn, not yarns.
As they are using a lot of American English, I wonder if 'cotton' as in cotton reel -
Is a reel some unit of cotton threads? I speak American English, but I have never heard that term. I would expect cotton thread to come in spools (even large industrial ones) or maybe skeins, if there were no object the thread were wound around. It's not something most Americans come across, though, in any case.