Why my Battery Protection + Charging + Buck/Boost design doesn't work
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Prior to design and mount the entire pcb, you must test every "module" in an isolated environment firstly. Don't believe your circuit will work as expected in the first try. It probably will never do. Hardware is not software in the real world: your theorically working circuit will crash in the real world in obscure and bizarre ways you will never expect. Cables are antennas, inductances, resistors and capacitors, every non-linear component will have high order components will make you suffer, digital and analog are enemies in the PCB...

If you say your DC/DC doesn't work, it means the DC/DC probably has a design error. So, the procedure to check correctly that module is:


Review your requirements. Your DC/DC has to be up to 3.3 volts? 5
Volts?
Isolate the DC/DC firstly
Analize the input prior to connect to the DC/DC. Is your input
correctly defined against your requirements? Good if so, check it
again if not.
Connect the DC/DC module without any other modules. Which voltage
is in the input? Is not the same as before, when you didn't connect
the module? If are equals, good. If not, your power supply will not
be acceptable to operate with the currents you are working-probably
you will have a shortcircuit somewhere
Check the output of the DC/DC module now. Which voltage are you
measuring? Its the same as the voltage you expect? If not, probably
your module will not have the correct feedback values.


At this point, you will need to check if all electrolitic capacitors are in good state and their '+' simbols are placed in the right place.

Checking the datasheet against your circuit, I observe your feedback resistors are not good.



Your R1 resistor in the sample (R9 in your schematic) is not the same value: 820kohm.

The datasheet does not contain high frequency filters-capacitors in the power lines lower than 10uF. Put a 100nF and a 100pF in parallel, too. It will filter high frequencies and, therefore, add more stability in the inductor when it creates the output voltage. The C3 and C4 in the sample circuits with two more smaller capacitors there, and another pair in the C1/C2 caps, too.

If, with all of these advices, your DC/DC does not work in the PCB with all the modules ready to be soldered, redesign in a separate module from scratch and test the PCB to ensure you will not have any short-circuits without soldering any component. Then, if your PCB does not have any short circuit, solder the components and measure the DC/DC nicely module you will have. If the DC/DC goes well at this point, and it works as expected, probably you will have a bad shielding (bad design of ground planes) in your circuit and you will need to isolate the power planes.