Does the A320 autopilot work in flight with both engines off?
You want to assume the APU is running. The FCOM says:

APU start is not available for 45 s after the loss of both engine generators. This 45 s delay prevents any interference with emergency generator coupling.

So the APU can be started following dual engine failure after waiting 45 seconds.
With the APU running, the green (G) and yellow (Y) hydraulic systems will also be powered, so AP1+2 should work. (The Y hyd will run via the elec pump on AC2, and with Y pressurized, so will the G via the power transfer unit.)
Descending by maintaining a speed using pitch control is called OPEN DES in an Airbus (FLCH for Boeings). OPEN DES doesn't require the A/THR to be working:

The engine-out descent strategy requires disconnection of the autothrust, and descent in OPEN DES mode.
Disconnecting the autothrust prevents an automatic setting of THR IDLE; therefore, the autopilot will fly the target speed in OP DES mode, with a thrust manually selected by the crew.
When reaching the FCU-selected altitude, or whenever normal descent is resumed to a lower altitude, reengage the autothrust.

The flight law reconfiguration table in the FCOM says the AP is lost when any of the items in the dashed box below fail:

I tried to disprove it, but it seems like with the APU started, an OPEN DES should work. (Report back please.)