Adapting delay circuit to 3v
Well, I got it.
First of, I opted for a quick working solution. This means I had to get it done today, with the parts I had on hand, and make do with a less than ideal result.

I changed to following values in the schematic you can see in the question. The result is a 2.5 minute delay before the LED turns off (last 20 seconds or so are pretty dim)
Power source is 2 AAA batteries (approx 3.2V)
2M2 resistor was replaced with two 100K resistors in series.
1000uF capacitor was replaced with 4 220uF capacitors in parallel.
I used one 2N2222 NPN transistor and one 2N3906 PNP transistor.
I took the LED off of the non working circuit. It's an unidentified white SMD LED. But it's super bright (enough for the night lamp) even at 10mA.
The LED is connected between the PNP emitter and ground without a resistor.