I would say codeine, dihydrocodeine, pethidine/meperidine, hydrocodone, and oxycodone are equally quick-acting.
Of the five, I've found hydrocodone to kick in the quickest, then oxycodone comes in second, then Demerol - but that's just me.
I think it will be different for everyone. But it's definitely one of the 5 I mentioned above when it comes to the oral route.
Demerol acts very quickly aswell orally. I know because I've used it through that route and (IV), and it kicks in orally just as quickly as hydrocodone or oxycodone do.
Morphine instant release is quick acting, as is Dilaudid and I'm sure Opana IR is also, though I've never tried the latter.
But there is a host of semi-synthetic opiates that that we will never get to try that probably act very quickly - heterocodeine, pentamorphone, hydromorphinol, and close morphine relatives like desomorphine, dipropanoylmorphine, and dibenzoylmorphine.