There are two verses that are relevant:
And We taught him (Dawud) the fashioning of coats of armor to protect you from your [enemy in] battle. So will you then be grateful? (21:80)
[Commanding him], "Make full coats of mail and calculate [precisely] the links, and work [all of you] righteousness. Indeed I, of what you do, am Seeing." (34:11)
The claim that Dawud (AS) was the first to make this type of armor is attributed to Qatadah. This claim appears to be supported by the verse as Allah first says "we taught him" then he ends with "will you then be grateful?" implying that this teaching to him is how we came to know as well.
Another thing to note is that many commentators have said that Dawud (AS) was given the ability to bend iron without needing fire and to make the armor very easily.
The issue is now two-fold: 1) when was Dawud (AS) and 2) when was this armor first invented according to history?
As for the first, any guesses are based on the Bible. That is not the most reliable source. But, it is not difficult to be sure that Dawud (AS) was long before the 5th century.
As for the second, it is definitely hard to know. The issue is of course that trying to know the non-existence of something historically is rather difficult. It is much easier to prove the existence of something rather than the non-existence.
To prove that something was invented at a certain point, one needs to prove that it existed after that point and did not exist before it. We certainly know that the Celts had this type of armor. But, do we know that it did not exist before them?
This is the point of doubt. Absence of evidence is not evidence of absence. It is a theory that it was invented by them but not proven fact.
It is not hard to imagine that Dawud (AS) was the first to make them and he was given ability to mold them easily, but they did not spread to widespread use other than Dawud (AS) because it was hard for others to manufacture them, then they resurfaced when people figured out how to manufacture them well.
And Allah knows best.