Janeway stated that only Captains (or anyone in command of a Federation Vessel) and Flag Officers have been briefed on it. Now let's say for the sake of it Janeway dies and Chakotay stumbles upon Omega. As a CMDR (or was he a LCDR) he wouldn't hold the rank or position to know what to do. I don't think the writers thought that threw or maybe I'm just overthinking it. Especially because Chakotay was never intended to "serve" on Voyager. Would Cavit know? hmm
The episode worded things a bit sloppily. Knowledge of the Omega molecule wouldn't just be limited to captains and flag officers, Janeway says that when Omega is discovered in the Alpha Quadrant a specialized team is sent in to deal with it. It's doubtful this team consists only of Captains and Admirals. Also, we need to consider starship captains who might hold a rank lower than Captain. More likely everyone Captain and above is briefed about Omega, as are officers of lower ranks who become starship captains. Although, anyone with a lower rank is doubtless informed if they are deemed need to know. Although, to answer your specific point, I guess had Janeway been lost and Chakotay assumed command of the ship, he likely wouldn't know anything about Omega, unless Janeway had an emergency packet prepared with info on Omega included within. Doubtful Cavit knew about it, unless he had history with these specialized teams that deal with Omega.
Doesn't Chakotay have his own codes? He'd probably been given then once he was made commander. I'd assume they would be upgraded to captain's codes once Janeway's death was officially recorded.
Well, yes, of course Chakotay has his own command codes for Voyager. But something like the Omega Directive is likely classified Captain's Eyes Only, and cut off from a Starfleet facility and an Admiral who could properly edit Chakotay's security clearance to allow him access to the Omega Directive, the computer wouldn't let him access to those files even after Janeway were lost, unless she had foresight somehow to grant him the access beforehand.
I think the opposite there. If Janeway were officially dead (and the computer knew it), then Chakotay would have access to the command files as the ship's captain, however some matters (like the omega particle) to a succeeding first officer might be need to know, at the computer's programmed discretion. Should the ship run across a omega particle, the computer would advise Chakotay that there was a classified file he needed to be briefed on in private. Chakotay would put someone in charge of the bridge, retire to his ready room and get briefed by the computer. Question is, would Chakotay track down and destroy the particle after the briefing with the same gusto Janeway did? Would he do it at all?