24V voltage regulator from ~55V/5A

Since you're going from ~18 volts to ~ 4 volts, this should more or less work as a voltage regulator. In this role,

1) Change C2 to 0.1 uF - the LM7815 specifies operation at this output capacitance, and 200 uF may well cause it to oscillate. Better yet, try just the LM1815 with the 200 uF cap (not the rest of the circuit) and see what happens. I presume you don't have a scope to look at the circuit, but at a minimum get a cheap DMM. Just be aware that any voltage you see that is not rock-steady is probably undergoing high-frequency weirdness that you can't see directly with a meter. Plus, if you do see what appears to be random drift, quickly touch the part. If you burn your fingertip, you know you need to turn off the power.

2) You don't say what the resistance of the pot is. Don't use 100 ohms.

3) Most important, get rid of C3. That large capacitor will almost certainly cause the output to oscillate.

4) A heatsink on the MOSFET is a very good idea. In theory, you can dissipate more than a watt in it, so you need a decent heat sink.

5) If you decide to do the prudent thing and actually test this circuit before you hook up to a battery, make sure you put a resistor (like, 10 to 100 ohms @ 1 watt) across the output.

With all that said, you need to rethink what you're doing. Recharging a NIMH cell with constant voltage is almost guaranteed to fail. Either the voltage will be too low, and you get no charge, or the voltage will be too high, and you kill the cell. Do some more research.

Let's have a look at datasheet Figure 10 "Dropout Voltage". At 500mA it will be a bit above 1.5V.

When operating in this mode, the output pass transistor will be fully on (saturated in this case, since it is a NPN bipolar).

If this was a PNP pass device LDO, you would expect excess ground current, as the regulator attemps to saturate its output device. However, this one uses a NPN, so the excess base current will simply go into the output. No problem here.

The regulator will not regulate anything though, this means it will act either as a resistor or as a couple of diodes in series, so output voltage may vary depending on current draw. Also, output voltage will follow input voltage, so input noise will not be suppressed.

If your load works on 19V, and you have 24V input, and the load can tolerate the regulator not rejecting input noise, then you're fine. In this case it would simply act as a voltage limiter.

If you also want to filter noise, then something like a capacitance multiplier with an output voltage limit would be more suitable.

EDIT: Example

^{simulate this circuit – Schematic created using CircuitLab}

This is a simple capacitance multiplier. It lowpass-filters the input (RC network) to filter out noise. Zener limits the voltage. I put in a CFP (double transistor) for lower output impedance, so you can say that's the "luxury" version!