ESP 01 lost its firmware when it disconnected
You have io 0 puled LOW so the esp-01 firmware starts in flashing mode and the firmware starts immediately after flashing (no reset). 

Remove io 0 to ground connection and add a pull-up for io 0 over a 10kOhm resistor if you want the esp-01 to start normally.

Note: powering the esp-01 from the 3.3 V pin of Uno (150mA max) could be insufficient for esp8266 WiFi operations at top transmit power (according to specs it can draw to 170 mA).