Divorce (talaaq) counts as such even if it is not registered in court

It is not essential for
divorce to be written down or recorded; divorce takes place as soon as the
words are uttered. So if a man said to his wife, “You are divorced” or he
says, if she is absent, “She is divorced” or “I have divorced her” and
similar phrases, then it counts as a divorce. It is not essential that it be
witnessed or that it be done in court or in front of the wife. 

Recording the divorce and
proving it in court is the correct action, so as to protect people’s rights
and prevent tampering with the limits set by Allah, especially if it is an
irrevocable divorce, such as if he has divorced her three times. In that
case, the sharee‘ah court has to be notified of that and issue
documentation, so that the woman who has been divorced can remarry if she
wants to, and so that the husband’s rights will be protected and the
divorced wife will not ask for maintenance after an irrevocable divorce. 

To sum up: divorce takes
place once it is uttered or written with the intention of divorce. As for
recording it and documenting it, this is to protect rights and is not a
condition of divorce taking place. 

And Allah knows best.