How much time is a commercial aircraft (e.g. A320 / B737NG families) flown manually?
Today's aircraft automation is just not set up for unattended operation.
There are too many individual tasks, that pilots must complete, to list here. The lowest workload for pilots is during cruise but even then ATC can instruct you to change speed, routing, or altitudes. Maybe one day this could be controlled from the ground but today only the pilots can make those changes.
I think you want to know how many minutes of an average flight are the pilots actually manipulating the controls either manually or through the A/P.

The A/P comes on somewhere between 30 seconds and 7 mins after takeoff and it stays on continually for the whole flight until maybe 2 mins before touchdown.
Every flight is different and sometimes there is lots of vectoring by ATC on departure/arrival and lots of step climb/descents, which requires the crew to &quot;fly&quot; the A/P. Sometimes the arrival/departure are linked up directly with the en route portion and you get a direct climb/descent.
So as a rough guess on an average flight the pilots might manually fly the aircraft for about 3–10 minutes and &quot;fly&quot; the A/P for about 5–10 mins. The rest of the time is spent talking to ATC, doing checklists, checking weather, fuel checks, and trying to keep yourself busy any other way you can think of. :)
I personally fly mostly ultra long haul flights of 13–15 hours as part of a 4-man crew so I also spend about 6–7 hours sleeping in the bunk.

In every phase of flight the crew is needed to do much more than just monitor the automation. The automation can not be used unless the pilots first enter all the required parameters, and then activate the automation at the right time, and in the correct order.
There is no aircraft that can take off on autopilot. The autopilot can only be engaged after takeoff. Once airborne at cruising altitude the aircraft could follow a programmed flight routing but any change in heading, routing or altitude must be manually entered by the pilot.
At the destination the approach must be manually programmed for the approach and runway in use. Once the aircraft in on the ILS approach an aircraft with autoland capability can land itself but it still needs the pilots to clear the runway.
Autopilot/autothrottle policies can vary with each airline. Typically airline pilots hand fly from takeoff to at least 200' but many hand fly to 10,000' or more for practice. When landing most pilots hand fly from about 1,000'.
Depending on the specific aircraft, autothrottle is usually engaged during the the takeoff roll and stays on until the flare just as it lands. There is no automation for landing gear and flap operation. It is 100% manual.