Arduino Uno and ESP8266 serial monitor issue
What you are doing wrong is:

If sending AT commands from PC to ESP and receiving OK ,as you said, works - that means you connected TX output from USB --> RS232 IC on the board to both RX pin of the ATmega328 and RX pin of the ESP8266 - not as in your picture. Similarly, you connected RX input of the USB --> RS232 IC on the board to both ATmega328 TX and ESP8266 TX pin of the ESP8266. If communication between PC and ESP works, then you connected TX Arduino pin with TX ESP pin and RX Arduino pin with RX ESP pin and not as shown in the picture.

When you send AT command from PC both ATmega328 and ESP8266 are getting the signal from USB --> RS232 IC in their RX pins. As your firmware in the first 10 seconds doesn't read the RS232 data, ATmega328 just ignores the packets sent by PC but ESP8266 receives the packets and responds with OK.

Since you shorted RX on ATmega328 and RX on ESP8266 and you shorted TX on ATmega328 and RX on ESP8266, ATmega328 can not send the data to ESP and vice versa so there is no OK.

In the other case, if you connected ATmega328 TX with ESP8266 RX and ATmega328 RX with ESP8266 TX (as shown in the picture) then you would have shorted USB --> RS232 IC's TX with ESP's TX and you would have shorted USB --> RS232 IC's RX with ESP's RX.

In that case ATmega328 might be able to send AT command to both ESP and PC at the same time and ESP would respond but now ESP's TX would be connected to ATmega's RX which is now shorted with USB --> RS232 IC's TX so PC can't receive that packet. However, when ATmega328 responds with OK, the signal now goes from ATmega328 TX to USB --> RS232 IC's RX (as always) and you might see the response in serial terminal.

So, you can't get the ESP to respond directly to PC and to ATmega328 without interchanging the TX/RX wires. Furthermore, as you may know RS232 is designed for communication between two devices, not three - and this is the case where three devices (USB --> RS232 IC, ATmega328 and ESP8266) are connected to the same TX/RX pins at the same time in a rather unusual way. Because, devices are usually connected TX to RX and RX to TX. However, when using the Arduino's onboard USB --> RS232 converter (by removing the microcontroller or holding it under reset) then you have to connect TX to TX and RX to RX. That is because there is one more TX/RX pair in between PC and the RX/TX pins on the board. The output of the USB --> RS232 converter (TX) sends the data to RX pin on ATmega328 (and RX pin on the board) but if you want to send the signal further (to some other device), then the pin marked with RX is going to act as TX.

To summarize:


The schematics does not correspond to what you have written. Such a connection would not allow to send the AT commands from the PC to ESP and to receive response, meaning you connected the TX/RX the other way round.
You can not have both at the same time. You have to decide whether you want to send AT commands from the PC or from microcontroller and then connect the wires accordingly.