CD4011BPWR Button Press Logic NAND IC
The Problem(s)

There are several issues with your CD4011-based design.


Digital chips do not have a well-defined switching thresholds. VIH and VIL set upper and lower limits, but the actual switching threshold varies, and is usually somewhere in between.
CMOS logic chips do not like slow-transitioning signals. They tend to draw excessive amounts of power when a signal slowly passes through the threshold.
Even if you used a chip with Schmitt trigger inputs like the CD4093B, both the high and low thresholds (VP and VN) as well as the hysteresis voltage (VH) vary significantly from chip to chip.
All logic thresholds change as the supply voltage changes. In this case, you can expect the supply voltage to vary between 4V (fully charged) and 2.9V (discharged).
A purely digital solution would be significantly more complex.




A Solution

This is one of the rare instances in which I would actually recommend using a 555 timer. However, you can't use a "generic" bipolar 555 (like the NE555), since it will not operate at a low enough voltage. Instead, you need a CMOS variant such as the MIC1555, TLC555, or LMC555, all of which can operate over a wide voltage range and consume much less power than a bipolar 555 (100μA vs. 5mA).

If you look at the schematic below, you will see the simplified internals of a CMOS 555 timer. First, note that the thresholds are defined to be at 1/3 (TRIG) and 2/3 (THRES) of the supply voltage, thanks to the three well-matched series resistors. This is convenient since the time constant of an RC circuit is (roughly) the time it takes to rise or decay to/from 2/3 of the input (battery) voltage.



There is also one critical detail that should be taken into account: TRIG overrides THRES, so it is preferable to use TRIG to detect a turn off condition. This will also prevent the circuit from automatically turning on during power up. In order to get the timing right, I used two RC circuits.


Turn-off: a 3.3μF capacitor (I recommend Tantalum due to its low leakage and temperature stability) in series with a 1MΩ resistor produces a time constant of ~3s, and decays to 1/3 of the supply voltage when the button is pressed.
Turn-on: a 100nF capacitor in series with a 1MΩ resistor produces a time constant of 100ms, enough to provide adequate adequate debouncing. This prevents the power from turning back on after you release the button. Note that a small N-channel MOSFET (such as the 2N7002 or BSS138) is required in order to invert the switch signal.


The output of the 555 is active-low in this case, and can directly drive your series pass transistor (the PMV30XPEA).



Here's an interactive simulation of the circuit.



I actually built the circuit using a TLC555, and it draws ~180μA over the full 3-15V supply range. I did make one minor alteration: the 1kΩ resistor was replaced with a 100kΩ to prevent the circuit from drawing excessive current when the button was pressed.