Hi! I recommend you to take a look to this similar project:
In this case, the ESP8266 is programmed and connected to the internet directly, without using the Arduino. The Arduino is only used for turning on and off the relays, or in your case, for reading sensor values.
So you should have to do two different Sketches:
Just a sketch that reads the sensor values an push them through the serial port. So the ESP8266 can read and save them in the variables that will be exposed to thinger. This is the most difficult part, as you need to encode the information in some way, maybe passing the numbers to string and writing each value in a single line
Connected to Arduino Serial port reading sensor readings. Then this values are parsed and stored in variables, just like they were read from the sensors.
Then when you are able to read the sensor measures from the serial port, it is easy to add the thinger code to make this parameters available to the platform. Notice that you should be able to program the firmware in the ESP8266. There are many tutorials out there to replace the AT commands with your own firmware, and you may need to use a USB To serial to program the ESP directly. You have here a small tutorial about installing the ESP libraries in the Arduino IDE, but you may need to find how to program it, as you need to put some GPIOs to GND. You can look here also as example.
This approach has several advantages: The ESP8266 is dedicated to the Internet and thinger connection, just receiving sensors data from the serial port and making them available. And the Arduino is only used for reading sensor values, forgetting the complexity of handling connections, internet connectivity, etc.
Please, share your progress and your problems. This kind of project is quite interesting taking into account that you are using a small ESP01!