Battery power back-uped circuit

[pedja089]

You shorted gate to GND in first schematic. So MOSFET are allways on.
As you see in document anodes of LEDs are connected to positive input supply via diode. Not to GND.
I overlooked D1 in second schematic. So my explanation for not powering load while charging is not correct.
But you must be careful with that schematic. As you can have 5.1V on output while charging. Then voltage will drop to 3.7V. You can't power modem from that. Not directly.
I'll suggest that you power modem directly from battery, and rest of circuit via 3V LDO, or 2.8V. You can get it witht Iq less than 1uA.
Then just build 1 or 2A 4.2V CC/CV charger for battery. I'll suggest that you use switch(step down), not linear charger.
If you want to stick to microchip AN1149, then you need really good SEPIC power supply for modem(to get recommended 3.8V). Also modem must have low ESR capacitors near it.
So not that trivial circuit to design.
EDIT:
SEPIC or some isolated topology is must, because you have higher input voltage when powering from AC, then output voltage(stepdown). But when you run it from battery you will have lower voltage on input then output.
You could try some hot swap PSU with battery while modem running, but i think modem will restart...