I built one of these in the 60s, put a microamp meter in series with the circuit.
Can it be done with LEDs?
Not with them as the only 'active' device. Their current-voltage curve is different than a gas discharge lamp, and they always consume some current at low voltage so the cap may never charge unless the resistor is small enough to also light the LED.
The tunnel diode approach looks good. A bipolar C-E transistor junction will conduct suddenly from voltage breakdown, perhaps not destructively at first. How about a Zener diode in series with the LED? Same issue perhaps, some current is always present and making C large enough could destroy the diode.
The beauties of the Nothing Box was the random operation and the long life of the battery. 9-Volt batteries would be practical today, but a stack of 15 of 'em is not cheap. The silver lining is to use the ones that are too worn out for whatever they previously went in.
Little thought was given to the brevity of the light pulse with the neon versions. The longer it is the brighter it should look. LEDs also have limits, more fussy than a tube about current because of the die wire.
The SCR is attractive or those who don't want ICs. Taking the SCR circuit as an example, the 2uF cap is big enough to put a resistor in series with the LED to extend the conduction, making the LED flash seem brighter to the eye than if it is over with in a few uS. On the other hand, reducing the LED current only makes it dimmer. 2uF@9V should be bright, but what is the peak current and is the discharge used in the best way visually?
Assuming T+RC to be 63% - at what point in the cycle is the SCR turned off by bias or threshold current?
Assuming a LED + SCR voltage drop of 2.6V
Assuming a LED to handle 2V@20mA (0.04W)
assuming the LED to flash 1 per second.
A decent compromise might be C=220uF and using 100 Ohms series resistance between LED and SCR.
The wires inside a LED are encapsulated in epoxy stuff and can't get rid of heat except by conduction to the die or the lead. For this reason, the current has to be limited. Conservatively, 3x the current should be OK for the wire at <30% duty. By experiment I ran a bunch of clear red and green LEDs at 5-8x the current until most failed. The dice failed, not the wire, on most I could see into. It can vary widely depending on the LED, just a suggestion. The place for improvement is probably the on time of the LED in a discharge-type flasher.
Assuming 1.2 Ohms circuit resistance:
The SCR circuit as-is with the 2uF cap looks like it would make flashes of light 1.5uS long at 5A peak current.
Putting 100 Ohms in series with the LED:125uS @ 64mA
Then making C 47uF: 2mS @ 64mA
Then making C 220uF: 14mS @ 64mA
Then making R 47 Ohms again: 6.5mS @ 136mA Which looks brighter? What's the storage time of the eye?
random stuff on Saturday AM.