How to communicate between a webcam and an Arduino UNO and get the live feed on mobile using ESP8266?
The Uno is a very limited platform, perhaps surprisingly, given its wide range of abilities and built-in hardware features. The two most glaring limitations for you application are its 16MHz clock speed (relative to modern desktop processors) and thus its bandwidth, and it's non-expandable 2KBytes of RAM.

Image processing would be right out the window, and even merely transmitting a raw image would be pushing its bandwidth limits, for any reasonable frame-size with even a very modest frame-rate.

Example: For a 100x100 8-bit frame at 15 frames/sec, a 16MHz processor (16M instructions per sec) would have (16000000 / (100x100x15)) = ~107 clocks (instructions) available to process each byte. You'd need some of those to either poll for next-byte-ready or to enter and exit an Interrupt Service Routine to process the byte. You're probably up to around 20 clocks (at a wild guess), to do nothing more than a bit of wire can do! Maybe you could do some trivial manipulation of each byte on the way through (say, brightness manipulation) with a few more clocks and you're already using a significant fraction of the processor just for that. And there isn't enough memory to be able to process a region of the image. And if you could, you run out of processor time before you could do much .... The problem just cascades as you approach the limits of speed and memory-space.