1. Prototypes
Status as of December 16, 2022
With a commonplace 16x2 LCD and a buzzer (on-board LED GP25 also used). LCDs are not well-suited for battery operation though (those with backlight), but they are comparatively cheap.
PCB Rev 3.1 w/ HV control loop (sort of final). The (chopped) humming is caused by the coil.
2. G-M tubes
In this project I use(d) two different G-M tube models: STS-5 and Z1A.
For registering pulses by the counting system proper working conditions have to be established. That means in practical terms generating a tube voltage within the Plateau area.
Beyond the "Knee" all pulses are (or should be) counted. Below the starting voltage, no pulses are counted at all.

SBM-20 (or STS-5, Cyrillic CTC-5)
As far as I know old USSR stock. Can detect only Beta and Gamma radiation (for Alpha you need expensive special design tubes). Technical data:
SBM-20_GER1.pdf (mikrocontroller.net)
For the tubes you are seeing on the pictures I paid around 30€, including shipment (bought one from a seller in Bulgaria, and one from an online store in Lithuania; one can also try to get cheaper ones in Ukraine, they are not so easy to obtain at the moment for apparent reasons).
Z1A
Origin unclear. Far less sensitive than SBM-20 (and probably most other counting tubes because of its small size), but okay for first tests and easily available. Likewise, can detect solely Beta and Gamma radiation.
3. Emitters
To test the general viability of the assembly I used small pieces of Uranium glass that can be bought in online shops (for obvious reasons this matter doesn't radiate intensely).
Natural radiation is detected about 20 CPM with the most recent prototypes and STS-5 tube (in 49°46' N, 11°12' E), but there's (inherent) randomness in the measured data.

4. Program development
C was the programming language of my choice.
Necessary for the HV-generation is a PWM with e.g. a frequency of 1 kHz and a duty cycle of 60% (output here on GP3, see next paragraph).
As we have the Pico MCU connected there are several ways to handle measurement data. Triggering IRQs is probably the most pragmatic way for a pulse (gas discharge, i.e. counting event) has a duration of about 0.3 ms.
A radiation source nearby can either be shown visually in a simple manner (flashing LEDs) or via clicks (buzzer), which are connected to one or more GPIOs.
For a more sophisticated display of data an off-the-shelf subassembly can be used. The Pico SDK makes common types of displays, e.g. a 16x2 LCD or a 0.96'' OLED, fairly easy to use - keeping in mind here that an LCD/OLED connected to the I2C bus may or may not have its own pull-up resistors.

5. Schematic
Diode D1 prevents immediate discharge of capacitor C1 and tube voltage is building up quickly. The coil voltage peaks into the double, then triple digits every time the transistor cuts off (see next paragraph).
Upper limit of the PWM frequency is a few kHz: around 2 kHz the HV is starting to drop considerably (see "Note 3").
The original minimalist configuration is now (PCB Rev. 3.x) endowed with a control loop for the tube HV.
Note 1: T2 MPSA42 changed to T2 MPSA44, see comment section
Note 2: values for R1, R5, C2, C3 can be varied
Note 3: tests suggest that UF4007 is superior to 1N4007 within the context of this design (UF4007 better suited for higher frequencies)

6. Simulation
Though a boost converter is neither a new concept nor very original (element values can be copied from other projects), it may be worthwhile to play around with a simulation (screenshots of the tool "MapleSim" below).
Note the exponential function build-up of the tube voltage.
7. EMC
Air wirings act like antennae and must be avoided for more advanced prototypes. Ground planes for PCBs are arguably advisable (minimizing EMI/RFI emissions), but they have to be properly designed (creeping...
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The counter on some of your photos is not SBM-20, but STS-5, an older version with similar specifications.