Initially I planned to find some cheap used joystick on flea market and just install switches instead of whatever it has on its axis to connect it as Sinclair joystick 1. But then I realized that there are still analog joysticks with DA-15 GamePort connector available even new on marketplaces so I found one for only $3.5. Ok, it took another $4 for delivery.. :)

These joysticks use analog potentiometers for X and Y axis and switches for buttons:

I decided to design an adapter to use the joystick as is and connect it as Kempson interface to appear on port 1Fh.

Circuit diagram of the adapter:

I used LM324 op-amp to translate analog potentiometers resistance to digital On/Off signal. Thresholds to switch Up/Down/Left/Right bits On or Off can be set using RV1 - RV4. 

Button signal must be inverted so I used 74LS00 for all needed logic here. Another option would be using buffer with inverted outputs in place of 74LS541 and set up op-amps network accordingly. I guess it would be less natural so it doesn't worth it.

Prototype of the PCB:

KiCAD project includes PCB design. I'm continuing to use the prototype, it works perfectly.