1. I only recognise the Oxford English Dictionary, not those upstart Americans ;-)

2. Even the Americans say (from your link):

"Irregardless was popularized in dialectal American speech in the early 20th century. Its increasingly widespread spoken use called it to the attention of usage commentators as early as 1927. The most frequently repeated remark about it is that "there is no such word." There is such a word, however. It is still used primarily in speech, although it can be found from time to time in edited prose. Its reputation has not risen over the years, and it is still a long way from general acceptance. Use regardless instead."

That's doesn't read as accepting the 'word' beyond the most technical noting that some people use it.

Dictionaries are historians of usage not legislators of language. At least in English, where we have no equivalent to the "Académie française" (suck it, Jonathan Swift).

Though it considers it "non-standard" the word also exists in the Oxford English Dictionary: https://www.oed.com/view/Entry/99668