The NodeMCU on its own, without the Uno, is by far the better solution. The NodeMCU is far more powerful than the Uno and using it as an AT Wi-Fi modem for the Uno is a bit like using a cart horse to tow a Ferrari.
Read this for more info:
I use a Wemos D1Mini with a couple of IR LEDs connected to control my aircon and fans. I don’t even bother with the resistors, so have just 3 wires plus the power connection.
You could probably do the same with the ESP-01, but I prefer the convenience of a dev board with an onboard debut port.
The issues you are facing with your current setup may be caused by the communication speed with the ESPs via SoftwareSerial, or poor coding, but without seeing your code it’s difficult to say.
Using breadboards for long-term projects isn’t the way to go. They are notoriously bad at maintaining good connections over time and tracking down the fault is a case of wiggling wires to figure-out where the dodgy contact is - not good for your sanity!
As I said earlier, you only really need two wires to drive your LED, plus an in-line resistor if that’s the way you want to go (the resistor value will be lower when using the ESP’s GPIO pins, as they operate on 3.3v logic levels rather than the 5v used by the Arduino).
Pete.