Yea, I think I realized to late that we are talking about outputs not inputs... but regardless... if you want TTL OUTPUT from an open drain/collector gate, you need a pullup.
Using ARDUINO DIGITAL PINS for OUTPUT into a NAND GATE, you would need neither PULLUP or PULL DOWN resistors. You would just directly drive the pins. It is the SWITCHES (having technically only 1 STATE) that need to rely on PULLUP or PULLDOWN.
In my case, I imagined (not yet seeing the missing drawing when I originally answered) that you wanted to feed the OUTPUT of the NAND to the arduino DIGITAL pin. In that case, you need to condition the voltage levels to match TTL using the method I described.
I now see that this is what you wanted... and it's that simple...
So, using NAND LOGIC... the LED ONLY lights up when PIN2 and PIN3 are both at logic "0". If any or both are LOGIC "1"... the LED is DARK.