Boost converter based on 555 timer not working
At about 5kHz your frequency is too low. Aim for about 50kHz. The circuit below has a timing capacitor of 1nF.

If you move your pot to the position shown below then you can vary the mark without altering the space.

7555 has minimum supply voltage of 2V. 7555 is the CMOS low power version of the NE555.

Diode should be a 1N5819 (Schottky diode) for fast switching and low dissipation because of its smaller forward voltage drop.

Also that's the symbol for p channel mosfet in your circuit diagram. See how the body diode would clamp the output voltage to 0.7V.