My choices for directors would be:
Rupert Wyatt
He did such a wonderful job with Rise of the Planet of the Apes. He's got what it takes to tell a great story, with characters you can care about, even when the main character is completely CG.
Matt Reeves
Reeves did a phenomenal job taking over after Reeves left, creating a compelling movie that was just as good if not better than the first. He could work wonders with Godzilla 2.
Guillermo Del Toro
This is an obvious choice, and probably very unlikely to happen since his plate is so full of projects. But I still think he can turn out an excellent Godzilla movie.
My first and only choice for composer is Michael Giacchino. He did an excellent job with Star Trek, blending the classic themes with the new. He could do the same with Godzilla. I also remember before Godzilla 2014 came out, there was a rumor that Giacchino was doing the score. In an interview he was asked if he was, and his response was, "I wish." So clearly he is a fan