I think I've managed to find the problem...
It's the dodgy hazard light switch, which never worked, so I've never used it. It won't stay in position, and it seems that the previous owner connected a stereo unit to it.
So basically,
1st position of the button - stereo is on, the battery light is glowing and I can switch left and right turn signal without ignition.
2nd position - stereo off, no battery light on the dashboard, no turn signal. Everything works as it should.
https://youtu.be/gg1HwAS-_fU
I've ordered a spare hazard light switch, so I'll give it try and see if this will help
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