Derbyshire, who has hosted a mid-morning show for a decade and prior to that hosted the breakfast slot, said she was moving to take up an unnamed "brilliant opportunity".
The trio are moving in what the station says is the "biggest shake-up to its daytime schedule in years", which will kick in from October.
Derbyshire - who was named speech broadcaster of the year at the Radio Academy Awards in May - said: "It's a huge wrench to leave 5 live where listeners and I have shared so many experiences, and I would like to thank them enormously for trusting me.
"As to what I'm doing next - well I've been offered a brilliant opportunity too good to turn down."
Afternoon presenter Bacon is leaving his show after seven years and among his future projects will be a BBC1 series hunting for amateur painters, which he will co-host with the actress Una Stubbs.
He said: "It's time to have different adventures, many of which will still be at the BBC."
Fogarty, a former breakfast co-host who now presents the lunchtime show, said she would be spending the summer "exploring a number of opportunities, before deciding what's next".
Adrian Chiles, who returned to the station last year after quitting the BBC to head to ITV, will front the 10am show on Mondays and Tuesdays.
Chiles, who was one of the launch presenters for the station, said: "Twenty years ago, I felt the luckiest man alive to land a job on the network when it launched. I feel no less delighted now to be part of this relaunch and I can't wait to get started."
On other weekdays Peter Allen will present the mid-morning show, moving from his position as co-host of the drivetime slot.
A new weekday 1pm to 4pm slot, called Afternoon Edition, will be presented by Dan Walker and Sarah Brett. And the 4pm to 7pm Drive programme will see regular co-host Anna Foster joined by former newspaper editor Tony Livesey.
Newcomers to the station include former Sky Sports News presenter Georgie Thompson and stand-up comic Josh Widdicombe who will join the team on weekend show Fighting Talk.
Station controller Jonathan Wall said: "This autumn feels the right time to refresh parts of our schedule as we follow-on from what's already proving to be a wonderful sporting summer."
"Victoria, Shelagh and Richard have all played such key roles delivering fantastic live radio and I wish them every success with their future projects."