General Discussion

There are very specific, and very detailed, rules governing discharge petitions.
Specifically, the procedure is as follows:
After the 218 signature threshold is reached, an additional seven legislative days must elapse before a Member who signed the petition may notify the House in a floor statement of an intention to offer the discharge motion on the floor. The motion may then be called up by that Member at a time or place, designated by the Speaker, in the legislative schedule within two legislative days after the day on which a Member whose signature appears thereon announces to the House an intention to offer the motion. The discharge motion is debatable for twenty minutes, equally divided between the proponents and an opponent and is not subject to a motion to table. If the motion to discharge a bill is adopted, it is in order to move that the House immediately consider the bill itself.

Thus, it will be at least a week before the discharge petition is brought up for a vote. One wild card is that a "legislative day" is not the same as a "calendar day" and I wouldn't put it past Johnson to muck around with the designation of a "legislative day" to delay a vote. In addition, my understanding is that the vote on the underlying bill is subject to ordinary parliamentary procedure, so a "motion to table" would be in order.

And, in the end, even if the bill passes, it has to go to the Senate and get 60 votes to be considered. And unless their is a seismic change in the repub caucus, that unlikely to happen. And in the event the bill does make it through the Senate, it can and almost certainly would be vetoed by Trump. Finding enough repubs that would vote to override that veto is yet another hurdle.

In the end, the likely benefit of the discharge procedure isn't going to be the release of the Epstein files. It's going to be forcing repubs in the House and, hopefully the Senate, to go on record opposing the release of the Epstein files.