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Both used to work with the nano. But when I tried to receive the correct libraries by loading them to the nano the display tuns out scrambled like in the picture below.
Ok. I see that the examples of this library produce output to Serial. So you could post this output from Serial Monitor.
And report which example you use.
And check and report the #define (s) for the pins CS, DC and RST you use for Wemos D1.
I use the library for an Adafruit display based on ST7735 driver. and as example I use the graphicstest.ino with next settings:
#include <Adafruit_GFX.h> // Core graphics library
#include <Adafruit_ST7735.h> // Hardware-specific library for ST7735
#include <Adafruit_ST7789.h> // Hardware-specific library for ST7789
#include <SPI.h>
#if defined(ARDUINO_FEATHER_ESP32) // Feather Huzzah32
#define TFT_CS 14
#define TFT_RST 15
#define TFT_DC 32
#elif defined(ESP8266)
#define TFT_CS D8
#define TFT_RST D4
#define TFT_DC D6
#else
// For the breakout board, you can use any 2 or 3 pins.
// These pins will also work for the 1.8" TFT shield.
#define TFT_CS 8
#define TFT_RST 9 // Or set to -1 and connect to Arduino RESET pin
#define TFT_DC 10
#endif
I expected you might have a boot mode issue, but your Serial output proves it is not a boot mode issue.
The execution times are nearly the same, so the SPI speed is the same most likely, not the cause of the issue.
The voltage level on the input side of the level converter could be a reason.
You could try to connect VCC to 3.3V. You might need to close jumper J1 then.
But you picture in post #4 looks different compared to the lcdwicki site.
Remains to question if the driver could be the cause, maybe the Adafruit library doesn't work with your display with ESP8266. You might try the LCDWICKY driver and example.
Your display has no SDO -> MISO pin, so it is write-only. You get garbage reading registers.
Last idea: search for ST7735 in this forum; there will be more to consider.