My uploaded sketch is running!
I have rebuilt the entire circuit on a different breadbaoard, with the same esp01-breadboard adaptor, but with the new usb/ttl adaptor. Plugged BarnabyBear's esp01 in, which ran fine as always. Plugged my other/old esp01 in, it did not run. Re-flashed the simple sketch from the IDE and... Bingo! It runs, I can see the hello messages on serial monitor. It runs fine powered by the 3.3V supply from my USBasp and runs fine using my new ams1117 regulator.
So, in my mind, the finger of blame points squarely at my old usb/ttl adaptor.
So I then moved the esp01 back to the original breadboard with the old usb/ttl adaptor. The sketch runs fine and I can see the hello messages on serial monitor. I then attempt to re-flash. But now, flashing fails every time with the message
warning: espcomm_sync failed
error: espcomm_open failed
If I move the esp01 back to the new breadboard/usb-ttl, I can flash OK and the sketch runs.
Finally I can continue developing with esp. Thanks so much to BarnabyBear and martinayotte for your patience and support.
@BarnabyBear, could I ask one more favour? If I post this old usb/ttl adaptor to you, could you test it to confirm my diagnosis?