Ius sanguini in French nationality law (or “droit du sang”) does not quite work in the way implied by some of your questions. Nobody can “get” citizenship by ius sanguini after the fact, you either already are French, or you aren't. If one of your parents is a French citizen at the time of your birth, you are yourself automatically and immediately a French citizen.
Proving your citizenship and enjoying it can sometimes be difficult in practice but no registration is needed. You can however get a foreign birth certificate transcribed at a French consulate to make paperwork easier down the line (which is somewhat akin to a “registration”) but you can also do the necessary formalities later in life. Legally speaking, you don't have to do anything special during your childhood to avoid losing your citizenship.
There is a catch, however: It's possible to effectively lose the benefit of French citizenship if it has not been used for two generations. Specifically, article 30-3 of the code civil reads:
Lorsqu'un individu réside ou a résidé habituellement à l'étranger, où les ascendants dont il tient par filiation la nationalité sont demeurés fixés pendant plus d'un demi-siècle, cet individu ne sera pas admis à faire la preuve qu'il a, par filiation, la nationalité française si lui-même et celui de ses père et mère qui a été susceptible de la lui transmettre n'ont pas eu la possession d'état de Français.
It's quite a lot to unpack. First, “possession d'état de Français” means “acting as, and being considered a French citizen by the authorities” (even if you aren't, incidentally). In practice, it means something like getting a passport, voting, etc. So what this article means is that someone who resides out of France and whose French parent also resided out of France for 50 years cannot reclaim their French citizenship if they have never used it.
Where this become quite complicated is that the 50-year delay and the lack of possession d'état are two separate conditions that must be evaluated independently (or at least that's my understanding). Previous court decisions also established that the 50-year delay is counted on the date of the request to have a person's French citizenship recognized (and not at the birth of the person).
Assuming your great-grandfather himself had possession d'état, if your grandmother is still alive, she could therefore presumably still apply for a certificat de nationalité française and then for a passport, ID card or something. At this point, she would have possession d'état and article 30-3 would not apply to your father, who could therefore also get his citizenship recognized.
On the other hand, if you cannot prove your great-grandfather had possession d'état (which is not the same thing as proving his citizenship) or if your grandmother did not have it and already passed away, then it's too late, your father cannot prove he is a French citizen anymore.
In practice, what you need to do is apply for a certificat de nationalité française from the Service de la nationalité des Français nés et établis hors de France. Given your situation, I would not be surprised if they would make difficulties so it might be necessary to hire a lawyer familiar with these issues to check if my understanding is correct and then navigate the process (which could, in the worse case scenario, necessitate an appeal to force the authorities to recognize your claim).
Even though I said that you cannot “get” citizenship, the “service” in question is seriously understaffed so actually getting your (theoretically pre-existing) French citizenship recognized can take a long time and in practice it feels a little like “applying” for it and waiting to get it. Legally however your grandmother, if she is successful, will be deemed to have been French from her birth, not from the date of the application, which is why she would have transmitted her citizenship to your father.
And then you would have to start the whole process again for your father and, if you wish, for yourself as you cannot start all three applications at the same time because you currently have no proof that your grandmother or your father ever had possession d'état. So it could take several years before all three of you have a certificat de nationalité. (Unfortunately, I cannot give any specific number, French citizens living abroad have been complaining about this for years but I am not aware of any hard stats on how long it takes on average.)
