This is a quick posting in case any one else gets tripped-up by the issue.
Although I have a number of Wemos D1 Mini devices working with Node-RED, this is the first time I've tried using the smaller ESP01s variety.
I recently flashed a set of ESP01s devices (with ESP Easy) and they all worked whilst they were in the USB Programmer, but when I mounted them on a breadboard - some of them wouldn't connect to my home WiFi. (i.e. the on-board small blue LED (that indicates WiFi connection) never came on.)
After a few hours pulling my hair out, I eventually tracked the problem down.
The ones that DO NOT have white silk screening on them need a pull-up resistor (10K ohms) from the CHPD pin (might be labelled as EN) to +3V3 (to enable the device otherwise it's disabled).