In the United Kingdom it is customary for a new peer or baronet to begin using the title following his predecessor's funeral, even if his succession has not yet been legally recognized. Is there a similar custom in Spain?
I don't know, but my understanding is that, in the UK, succession to a peerage is in fact automatic, meaning that the title doesn't go vacant when the last holder is deceased and there is a living successor. Having the successor's name entered in the Roll of the Peerage is a condition not for succession properly, but rather to legally enjoy the rights, privileges and precedence of the peerage. A peer whose name is not entered in the roll is still a peer, but cannot, for example, use his title or be cited by his title in official documents.
In Spain, on the other hand, the title remains legally vacant (unlike in the UK) until a
Real Carta de Sucesión is issued in the King's name by ministerial order published in the State Gazette. To have such letter issued, the person who believes to be the successor has to file a petition with the Ministry of Justice and pay the appropriate succession tax .
Usually, when a person is the immediate successor under the regular order (since 2006, according to equal primogeniture), that should be a straightforward process, but that is not the case of Tamara Falcó.
Since her father expressed a desire in his will to distribute his titles among his descendants, reserving the main title to the firstborn, it is unlikely that Tamara's siblings will contest her succession, but,
if I understand it correct, Spanish nobiliary law allows for a period of time during which the siblings (or any person who believes him/herself to be of better right) could do so. Is that perhaps the reason why the process takes so long? Tamara's father passed away early this year and the title is still vacant today.