Have you tried Merriam-Webster? The American Heritage Dictionary? Dictionary.com (based on Random House)? These all list “fast” as an adverbial form of “fast”, meaning swift. And when I said the Oxford English Dictionary cites went back to 1205, I didn’t mean they were only from the thirteenth century; I meant they documented an unbroken chain of usage from then up to modern times.
Oh, and, unfortunately, the reality of the modern educational climate when it comes to linguistic matters is such that “years of teachers correcting students” isn’t a very reliable indicator of the actual facts of the English language.
Given that you yourself admit that “fast” when used this way is ubiquitous among native speakers, to the point that most do not consider it jarring or malformed, even on sober reflection, it is rather silly to think that it is still not really part of the language proper.
As it happens, according to the OED, the word “fasten” is etymologically related to the word “fast” meaning “of great speed”. In fact, the use of “fast” to mean “of great speed” actually historically developed out of a primary usage of “fast” to mean “firmly fixed”. Actually, the shift from “fast” meaning “firmly fixed” to meaning “of great speed” occurred first in the adverbial sense of “fast”, the shift later transferring to the adjectival sense as well.
Quick links to the relevant dictionary entries on “fast” as an adverb dealing with speed:
Oxford English Dictionary (sense 6a; note the string of cites running continuously from the beginning of the 13th century right up into the 20th century). Observe also that the corresponding meaning for the adjective (sense 8a here) doesn’t really start until the middle of the 16th century.
Merriam-Webster (sense 3a)
American Heritage Dictionary (sense 3 of the adverb; note that it is presented with no Usage Note)
dictionary.com (sense 18 of the first entry)
Next
Firstly, blah blah blah…
Nextly; …
(I do actually try to use *nextly *sometimes, but only to annoy my children, who always pick me up on it)
Tell your children they have much to learn and Daddy(or Mummy) knows best.
From the Oxford English Dictionary:
Cites range from 1572 to 1974.
Well I never! Thanks for that.
I think it must have fallen out of use, because it does sound awkward (which alone is enough reason for me to continue using it).
Fast is the adverb for the verb fast.
Alright then, now you’ve got some citations, so I sit corrected. Thank you.
One time in the Great Lakes region I came fast on an obstacle, and people recommended Niagara.
I’m no grammarian, But why wouldn’t timely be an adverb of time?
“He arrived in a timely manner.”
Maybe that’s more of an adverb for opportune. Hrmm.
That’s an adjective. It’s modifying “manner”, not “arrived”.
Apace is an elegant old word that means the same as rapidly, even though it doesn’t seem to look like an adverb. “The car moved apace toward Annabelle’s cottage.”
This use of “fast” probably is another meaning of “fast.”
“I have come fast on an obstacle,” means “I am stuck on this rock.”
There’s “basic” and “basically.” WTF happened to “basical”?
Was it a buoy or a gull?
Speedily, anyone?
The modifiers of modifiers often don’t end in -ly. For example, “very”, or “quite”.
And I’ll add my voice to the choir that “quickly” is the adverb corresponding to “quick”, “rapidly” is the adverb corresponding to “rapid”, and “fast” is the adverb corresponding to "fast.
He arrived timelyly? 
Think about that, though. “He arrived timely” just sounds wrong. “He arrived quickly”, “he arrived hastily”, “he arrived speedily”, yes. Those work. But he arrived timely?
Most of the “ly” adverbs are made by adding the suffix to adjectives. “Timely” is not constructed that way. “Time” is a noun, in this context. So “timely” doesn’t work in adverbial constructions.
Whoops! Thought you were the person who originally posted the sentence I critiqued, severus, responding to that critique. Never mind!
No problem. This said, is there an adverb formed with “timely”?