You want to pull the 555 control pin low, then raise it gradually. The 555 has an internal network of 5K resistors. To overcome their effect, your device needs a low impedance.
You have a slowly charging capacitor which will provide the control signal. However its charging resistor is high impedance.
You need something to go in between the two.
A pair of transistors may do the trick.
Schematic:
You may want to alter the output's beginning and end levels. This may be possible with another resistor or two.
The timeframe is 10 seconds using a .1 uF capacitor. By substituting a 10 uF, it will extend the timeframe to 1,000 seconds.
Not shown is a way to discharge the capacitor. This will take some more work. It may turn out to be easier to charge the capacitor at the beginning, then let it discharge.