555N vs. 555P? One works, the other does not!
In general, when a circuit works with a device from one maker and not from another it is telling you is that your design is incorrectly using some feature of the device on the verge of the design characteristics. 

In this case how you are driving the reset pin.

That is a bad thing. You say it works with a 555N, but I am willing to bet if you tested enough 555N samples, only a proportion of them will work. So be glad you tried the other part.

Fix your design so the reset pin gets what it really needs, i.e. a digital level not an analog voltage that hovers somewhere between the logic thresholds as defined in the worst case 555N/555P data-sheet.

You likely need to add a simple comparator circuit, with hysteresis, to do that.