will this timer/delay circuit works as intended with almost equal opamp inputs?
the author confirms that only two terminals of the pot are used in the
  comments.


Then he's stupid. 

This will not work as I see it working. You need to set a defined voltage at +Vin by using a potential divider and not rely on the giga ohm input impedance of the op-amp's input resistance as some kind of valid value. 

Other problems: -


The TL082 minimum recommended supply is 10 volts and not 9 volts.
Input common mode range is guaranteed to be +/- 11 volts on a +/- 15 volt rail and if your supply rail is (say) +10 volts then the input range is from +4 volts to +6 volts.
Output voltage range is guaranteed to be +/- 12 volts on a +/- 15 volt rail hence, on a + 10 volt supply, the output cannot be relied upon outside the range +3 volts to + 7 volts i.e. your LED will likely always be on. Typical limits might cause the LED to switch off but you shouldn't rely on typical limits because you'll get bitten.