I'm sure I have found the reason why the ESP-01 wouldn't flash on the Witty base, whereas the ESP-07 will.
The ESP-07 breakout board comes with a 10k pull-up resistor to CH_PD (EN) as well as a 10k pull-down to GPIO15. Now, ESP-01 doesn't use GPIO15, but certainly uses CH_PD, like all ESPs do!
The little USB stick thing that I use for programming ESP-01s obviously has this built in already but I am guessing that the Witty base does not, because it lives on the top board, just as it does for the ESP-07 breakout. Which explains why I could program my ESP-01s on the USB sticks but not on the Witty base. (Although note that you still do have to pull down GPIO0 when flashing the ESP-01 on the USB stick - not so on the Witty base, of course, as that does the DTR/RTS trick for you automatically)
The above dawned on me when I found that even though my ESP-01s worked perfectly when run off the USB-stick, they would not work when removed and run standalone. The addition of a 10k between CH_PD and 3v3 solved that problem!
I am practically certain that was the cause! I haven't got time to test it right now, but I will soon be making a "breakout-breakout" board for the ESP-01, to allow it to be plugged in to the Witty base. The USB stick, even though it works great, is quite fiddly, whereas the Witty base is extremely handy for me (ymmv).
Another thing to remember is that on the ESP-01 you have to cross the TX and RX lines, they are not straight through connections as in the case of the ESP-07 form factor.