I'm going to take a shot at this, given no one else has bothered. It's not an area where I have much experience. But I do have a little bit. So I'll draw from that.
The Cockroft-Walton multiplier you are considering isn't the direction I'd go. But at least I understand why you are trying to use a \$24V\$ rail, instead of the \$12V\$ output of the 555. You use about half as many stages that way. But assuming you use a complete, well-designed push-pull stage at the end of the 555, you still will need something like 7 stages (one of them to cover the diode losses and your push-pull losses.) Not the 5 that you show. I think 7 may get you there. Also, your push-pull driver will have to source and sink some serious current as it charges up the capacitor chain. You may want to plan on \$5A-6A\$ peak in the early period, sink and source, for designing purposes -- though once everything is fired up it will be a lot less. Ripple at 5mA load is another issue to consider. I'm guessing each capacitor on the order of \$1\mu F\$ in the multiplier chain. Just thinking about the 7X staging more, I'm guessing that you will need about \$70mA\$ for perfect efficiency (\$7\cdot 2\cdot 5mA\$.) But due to losses then probably more like \$100mA-120mA\$ rms. Peak current, worse, of course. That's where I'd put my mind at before building something like this and finding out, anyway. Frequency will also matter a lot. For the above thinking about capacitor sizing, I just picked a \$200kHz\$ drive frequency for consideration. But that itself will take some care with the push-pull design. It's doable. But not trivially doable.
(I'm more comfortable driving the push-pull with BJTs and perhaps using some speed-up caps with a different topology for your \$Q_1\$ section. But that's me.)
A boost topology just seems more achievable to me. I'd start with the following equations and attempt first to consider just a direct drive with the 555 at \$12V\$:
\$V_{in} = 12V, V_{out} = 144V, I_{out} = 5mA, V_{fwd} = 400mV, V_{sw} = 200mV\$
\$V_{on} = V_{in} -V_{sw} = 12V - 200mV = 11.8V\$
\$V_{off} = V_{out} + V_{fwd} - V_{in} = 144V + 400mV - 12V = 132.4V\$
\$D_{on} = \frac{V_{off}}{V_{on}+V_{off}} \approx 91.8\%\$
\$I_{peak} = 2\frac{I_{out}}{1-D_{on}} \approx 122mA\$
This leaves the inductor to design. One fact follows from the basic inductor equation of \$V = L \frac{dI}{dt}\$: namely, \$f\cdot L \le \left( \frac{V_{on} \cdot D_{on}}{I_{peak}} \approx 88.79 \Omega\right)\$, where \$f\$ is the frequency of operation. I'm going to select \$f= 200 kHz\$, arbitrarily, because it is achievable and will help reduce the inductance I require. From this, I estimate that \$L \approx 444\mu H\$.
Let's select the Amidon FT-50-61, which should be okay for a frequency I hope to be able to achieve (it works well from about \$200 kHz\$ and up.) From the datasheet for it, I find that I need about 80 turns to achieve the desired inductance. Using #28 wire, it's about \$\frac{1}{4}\Omega\$ of resistance in the 5 ft, roughly, of wire... nice.
Ferrite should use \$B_{max} \le 0.1T\$ It turns out that: \$f \ge \frac{V_{on}\cdot D_{on}}{B_{max}\cdot A_C \cdot N}\$, where \$N\$ is the number of turns and \$A_C\$ is the cross-section area of the core (toroid, perhaps?) Using \$B_{max} = 0.1T\$ and a cross section of \$A_C = 0.135975 cm^2\$ and, of course, \$N=80\$, I get \$f \ge 99.6 kHz\$. Lucky for me, \$200 kHz\$ is more. So, it looks okay on that score, too. (That equation took into account the number of Webers needed.)
Now at this point, I'm worried about the high ON period. But a quick check of another equation that suggests the maximum duty cycle seems to suggest this is okay: \$D_{{on}_{max}} \le \frac{V_{on}-I_{out}\left(2 R_L\cdot R_{on}\right)}{V_{on}+I_{out}\cdot R_{on}}\$. \$R_L\$ is the inductor's resistance and is already known to be pretty low. \$R_{on}\$ is the switch (transistor) resistance. And that is also pretty low, even if using a BJT rather than a MOSFET. Plugging in values I find that \$91.8\%\$ seems okay.
So. That's maybe a first cut on a design to breadboard and test. Just get some #28 magnet wire, one of those FT-50-61 toroids, and wind it. You can go through these steps with your \$24V\$ source if you want, too. Either way, you may get something that works.
Keep in mind that the above design assumes you are filtering at the output to smooth things out. I suspect you can work out those details. It also assumes your load is actually \$5mA\$. If you want to design it for more, feel free. I gather you'll regulate the voltage, anyway, so I suppose you can leave the oscillator running at a fixed frequency and let the regulator dump energy as needed to keep the output voltage where it needs to be.
However, consider something like this in the following link:
Small, High-Voltage Boost Converters
I didn't spend much time reading through it, but it looks roughly like something that may apply. (It may be nearly useless at your low power level, as I didn't check to see.)
