It is in the tools menu where you selected the Nano, it should give you a choice of which boatloader to use. I cannot say for sure it depends on the IDE you are using.
Many, many thanks, sterretje.
I did as you said, press reset and the LED flashed quickly. So it seems that's the new bootloader. Next was setting to ATmega328P, and o wonder, the upload functions!
Problem solved, I think.
The other points are irrelevant, the Nano is connected to nothing atm.
I just checked another, older Nano, this did show only a single flash, so uses the old bootloader (also the USB connector is not the same type). Not sure if that shows something important, and I don't know the names of all these connectors.
So, at least I know now that I use a clone with the new bootloader - just a bit surprised that it takes so long to solve that mystery. But in the end the result counts, and for that I'm really thankful.