Electrical – Discharge Characteristics of Lithium Ion Battery

This seems to be very similar to Nick's schematic, was probably busy drawing it when he posted :-).

First why you can't use the N-FET on the high side: it needs a gate voltage a few volts higher than the source, and the 4.2 V is all you have, nothing higher, so that won't work.

I have a higher value for the pull-up, though a value of 100 kΩ also will do. 10 kΩ will cause an unnecessary extra current of 400 µA when you're measuring. Not the end of the world, but it's 1 resistor in both cases, so why not use a higher value.

For the MOSFETs, there are a variety of parts to choose from given the requirements are not so strict; you can consider inexpensive ones such as, e.g., Si2303 for the P-channel and BSS138 for the N-channel.

Batteries are tricky nonlinear devices. If you want to build a safe and reliable lithium battery charger, you need to know more about your batteries and your battery charger circuit. In general, lithium batteries are not interchangeable and not all chargers will work with all batteries.

There are also significant differences between lithium (probably not what you mean), lithium ion (probably what you mean, sometimes written Li+ or Li-ion), and lithium polymer (sometimes written LiPo) batteries, and significant differences within the battery chemistries of these categories.

The datasheet I found for TP4056 does not say that this device includes undervoltage protection. It only provides charging. Perhaps other circuitry does on that evaluation board does. The appropriate discharge protection is a function of battery chemistry, and the threshold should set according to the battery manufacturer's datasheet (as a baseline, anyway).

A circuit that tests for undervoltage is probably measuring the battery under load, and will need to compensate for the battery's internal resistance \$R_{internal}\$ reducing the voltage at the terminals \$V_t\$. That is

$$ V_t = V_{oc} - IR_{internal} $$

So to answer your second question it's quite possible that \$V_t\$ = 2.5V is an appropriate cutoff for a battery with \$V_{oc(min)}\$ = 3.0V, if \$ IR_{internal} \$ ~ 0.5V.