Actually, it's pretty simple. Gate the Qx output with the outputs of Q0-Q3 in various combinations to achieve the delayed output signal. One way to do this in an adjustable manner is with a 4-bit magnitude comparator (7485 or equiv.), a flip flop, and either a 4-bit DIP switch or a small binary-encoded rotary switch. 4 bits gets you delays of all values from 0 to 15 clocks.
And, again, you have only discussed delaying the leading edge of the Qx output waveform. What about its trailing edge?
ak