The 20,000-square-metre warehouse - which is also home to music distributor PIAS - was set on fire during a night of riots in Greater London.
A statement sent out by PIAS late on Tuesday evening read: "Sony DADC have actioned their Business Continuity Plan and are back up and running from a new control room in Enfield. PIAS continue to work with them to minimise the impact on the business, a number one priority for all labels and clients.
"Sony DADC have identified a temporary distribution partner and it is envisaged that they will be in a position to pick, pack and ship orders in the course of next week."
The company promised a further update on Wednesday morning, adding: "We wish to thank everyone for the overwhelming messages of support."
PIAS handles releases from a large number of independent record labels including XL Recordings, Domino Records, Beggars Group, Wall of Sound, Sunday Best and Memphis Industries.
A number of independent labels have confirmed that some or all of their stock was lost in the fire, while a fundraising effort for affected artists and labels has been set up under the name 'Label Love'.