I recently bought a new ESP8266 because I'm pretty sure I used 5V instead of 3.3V on my old one.
But it seems the new one behaves the same as the old one.
When I open up the serial monitor and type AT I get not response (on the 115200 baud).
If I boot up the power supply while the device is connected I get this in the serial monitor:
And of course, you removed the ATmega328 from the UNO, didn't you?
Look, this has been the basis for much discussion over time. The diagram (not sketch) you show is incorrect as it puts 5 V on the RxIn pin of the ESP01 - not that this would be the problem you describe here. This can be corrected by putting a diode with anode to the RxIn pin and cathode to the Arduino, but I seriously suggest you do not use this lash-up and wait until you can get one of these from eBay:
Actually, get two (or more, if you are going to use a number of ESP-01s).
These can easily be modified to add a programming jumper and interface the ESP8266 to the USB port perfectly, so they do most of whatever you would want to do with an ESP-01. I will be using one to drive a "Times Square Display", plugged into a "Phone Charger" as the power supply.
It is otherwise a waste of time connecting an ESP8266 to an Arduino system as the ESP8266 alone, being a substantially more capable processor, will do pretty much everything you need, you may just have to use one with more connectivity such as a WeMOS D1 Mini or NodeMCU.