Misunderstanding document about leakage inductance in a flyback converter
So if the current through Lp is not null, as the current to the
  secondary is proportionnal to the current through Lp, it is not null


The current in the secondary is NOT proportional to the current in the primary; the voltage on the secondary is  proportional to the voltage on the primary. Of course any secondary current modifies this but, until the primary voltage has risen above the bus voltage (Vin), even an ideal diode in the secondary won’t conduct.


  I do not understand why the current through the diode is null for a
  certain time.


Due to the parasitic capacitance lumped at the drain, there has to be some time before the primary voltage rises to be bigger than Vin.