As I see, you have some confusion with the connection. Signals SDA & SCL refer to i2c bus, but Uno pins 10-13 - to SPI bus.
Do your sensor has I2c or SPI interface?
1. The connections to the BMP 280 as shown in your photo within post #13 really need to be soldered to the board. As you will never be able to guarantee a reliable connection by just poking the pins into the via holes of the BME280!!!
2. The Arduino connections shown in your photo within post #8 A5 & A4 are not the Arduino Uno r3's SDA & SCL (I2C) connections - If you look at the Pinout Diagram shown for the Arduino r3 you will see that the I2C port resides on pins D19 (SCL) & D18(SDA)
You need to have reliable connections to the correct port or this never going to work!
This are the same pins. If you look at the schematic on your own link -
you will see that pins A5 A4 on Uno are directly connected to pins D19 D18.
Moreover, on the Nano there are no separate pins for SDA and SCL, and I2c bus is located on the pins A4 A5