The code is in his first post. After a lot of cross checking it appears that it is probably wired correctly. I'm not 100% positive since the angle makes it hard to tell which pin on the breadboard lines up with the ESP module.
Buzzy, do you have any LEDs that you can include in your circuit? Like put one (led and series resistor) on the input of the shift register that you have the switch on. Another led on the clock line, and another on the MISO line.
One thing you could try is to disconnect the line from the shift register output and connect it to Vcc. Put a high on the MISO line of the ESP. You should then get what looks like all highs shifted in. Confirm this with your serial.print output.
It is not apparent what your problem is at this stage so you need to do some troubleshooting. Break it down to smaller pieces.
And while it may not be the problem, do not assume that your breadboard is making a good connections. This is why I would put lights on the board. You need to see what is going on. If you had any tools (scope, logic analyzer) that is what you should use. But if you don't (I assume) then you need to slow things down and confirm that the circuit works. And I would do this by using programmed I/O with long delays rather than the normal shift function you are using.