I think 'not sooner than' is a right form and 'no sooner than' seens pretty odd to me. Does anyone know where the phrase 'no sooner than' came from? Is it just a tradition?
"Sooner" is a comparative adverb. You use it when you are not placing two things before or after one another but comparing their timing. It is not correct to say "not sooner than" because a comparison is not binary, unlike situations where something either is or is not. There may be a whole range of specific times down to the millisecond between two points. The phrase you may use instead if you want to say something must not occur until a specific time is "not before".
You may open this gift no sooner than December 25th.
You may open this gift but not before December 25th.
It isn't just "sooner" that you use this way:
The oven should be no hotter than 180 degrees.
The oven temperature should not be above 180 degrees.
The structure no sooner...than can be used If the second event occurs immediately after the first, and it introduces the event that occurred first. Beginning a sentence with the negative word no, the structure requires putting the auxiliary verb before the subject:
No sooner had I arrived/did I arrive at the station than the train came.
This means that I arrived first and the train right after me.