routerRX -> LogicLevelConvertorTX1 -> TX0 -> ArduinoTX(pin1)
routerTX -> LogicLevelConvertorRX0 -> RX1 -> ArduinoRX(pin0)
I think you have some directions backward.
First of all those are "I" and "O" for Input and Output, not 1 and 0 for One and Zero.
TXI (Transmit Input) is on the LV side (of both channels.)
RXO (Receive Output) is on the LV side (of both channels.)
You want to connect the 3.3v TX of the Router to a TXI input and the corresponding 5v TXO to Arduino RX.
You want to connect the 3.3v RX of the Router to an RXO output and the corresponding 5v RXI to the Arduino TX.
routerTX -> (TXI>LogicLevelConvertor>TX0) -> ArduinoRX(pin0)
routerRX <- (RXO<LogicLevelConvertor<RXI) <- ArduinoTX(pin1)