It appears that Peyton Manning was not actually "waived" but "released".
According to the NFL CBA, once a player has completed his fourth year in the league, he is no longer eligible to be placed on waivers and is instead released as an unrestricted free-agent and can be signed by any club at any time.
Whenever a player who has finished the season in which his fourth year of credited service has been earned under the Bert Bell/Pete Rozelle Plan is placed on waivers between February 1 and the trading deadline, his contract will be considered terminated and the player will be completely free at any time thereafter to negotiate and sign a Player Contract with any Club[...]
As far as what it takes to make a "waive" or "release" official:
- First, the team must notify the player:
Section 4. Notice of Termination: The Notice of Termination form attached hereto as Appendix H will be used by all Clubs. If possible, the Notice of Termination will be
personally delivered to the player prior to his departure from the team. If the Notice of
Termination has not been personally delivered to the player prior to his departure from
the team, the Notice of Termination will be sent to him by certified mail at his last address on file with the Club.
- Then, the team must notify the NFL and the NFL must notify the NFLPA by noon the next day:
The NFL shall inform the NFLPA of player personnel transactions communicated in the Personnel Notice between the NFL and its member Clubs concerning the termination or trading of players including awards on waivers, termination through waivers, confirmation of trades or any change in the status of players (e.g., placed on Reselve Injured, etc.). The NFL will make best efforts to communicate the information referred to in this Article to the NFLPA on the same day, but in no event later than noon on the next day.
It is my assumption that the "waiver wire" that is referred to by Adam Schefter and other NFL writers is just their term for the communication that happens between NFL and the NFLPA, et al. Which is to say, the communication from the NFL that the player has been cut. So, if the Indianapolis Colts have already filed the appropriate paperwork with Peyton Manning and the appropriate league offices, the latest that he should be available to be contacted by other teams would be tomorrow at noon.