Has "aught" survived in common usage?
In a movie that I watched recently, I heard-
- for aught I know,
- for aught I care.
I work with a lot of native speakers, and they all told me it's not in formal or informal usage anymore.
What's the ground reality?
Yes, it survived, but it is commonly spelled owt.
The original spelling of aught, meaning anything, as used in the OP's question, is archaic, (Dictionaries: Oxford, Chambers).
An alternative meaning is "zero" (derived from nought.) This meaning is used when naming rifle cartridge sizes, and in Norfolk. citation needed, and see @WS2 's answer.
Both meanings under discussion (anything and zero) are alternatively spelled ought. (Oxford dictionary: ought-3 aught-2. Not in Chambers)
It remains current in Northern English with the newer, apparently 19th century, spelling of owt (Again in Oxford but not Chambers.) There it retains the meaning of "anything", opposite of nowt ("nothing".) As you can imagine, the pronunciation is slightly different.
For example, you'll find it in the so-called Yorkshire motto:
"‘Ear all, see all, say nowt. Eat all, sup all, pay nowt.
And if ever thou does owt fer nowt – allus do it fer thissen" BBC America
You can hear it used regularly on British drama series such as Coronation Street and Emmerdale, and the long-running sitcom Last of the Summer Wine (thanks @Pharap ) though I'll leave it to others to source quotations for you, but you can infer that it is understood by their audiences of millions of native speakers.
Here, Yorkshire poet Ian McMillan uses owt in a 2014 article in a regional newspaper. You can see by the way he uses it that Northern dialect is more often spoken and informal. ''Grappling to get to grips with an alien language'' by Ian McMillan
Update: According to the Dictionary of the Scots Language (DSL), there are also current Scots, Scottish English and Orcadian versions of aught (anything): ocht, ochts, oucht, owt, och. DSL has several quotations from the 20th century and earlier. Surprisingly to me, it lists four further meanings listed: a somebody, either, any, somewhat.
Second update Chambers and Oxford dictionaries trace the "anything" meaning of aught / owt to Old English awiht meaning "any person." (Bosworth-Toller Anglo-Saxon Dictionary entry)
I have added above that the alternative spelling of ought means both "zero" and "anything".
The ground reality is that a few well educated speakers will understand what this word means, most native speakers will be able to correctly guess the word's meaning from context, but very few people will ever actually use it in formal or informal speech.
However, "for naught" appears in the idiom: "It was all for naught", which is more commonly used.