No one, at least initially
One reason that both Tarkin and Krennic appear to be "in charge" roughly contemporaneously is that Palpatine has an actual policy of letting no single person control all the Death Star’s development:
All the same, he had no compelling reason to solve the enigma of the
battle station’s beginnings. What bothered him was that, compliant
with a strategy that no base commander—Moff, admiral, or
general—should have unrestricted access to information regarding
shipments, scheduling, or construction progress, no single person was
in charge of the project, unless of course the Emperor was considered
to be that person.
Tarkin
That said, descriptions of Tarkin’s role in the project from Tarkin and Catalyst broadly align:
And there were rumors to the effect that Tarkin had been instrumental
in convincing Palpatine to move forward with the battle station—even
that he had had a similar weapon in mind even before the discovery of
the Geonosian schematics. Serving nominally as adjutant general for
the navy, he was rapidly being brought up to speed on the status of
the battle station, and there was talk among the members of the inner
circle that Tarkin was being groomed to assume leadership of the
entire project.
Catalyst: A Rogue One Novel
This is broadly consistent with the narrative presented in Tarkin:
If establishing the identity of Sentinel’s enemies was already proving
daunting, getting to the truth of the battle station’s origin was
nearly impossible. Everyone from celebrated ship designers to gifted
engineers wanted to take credit for the superweapon. Tarkin himself
had discussed the need for such a weapon with the Emperor long before
the end of the Clone Wars. But no one outside the Emperor knew the
full history of the moonlet-sized project. Some claimed that it had
begun as a Separatist weapon designed by Geonosian Archduke Poggle the
Lesser’s hive colony for Count Dooku and the Confederacy of
Independent Systems. But if that was the case, the plans had to have
somehow fallen into Republic hands before the Clone Wars ended,
because the weapon’s spherical shell and laser-focusing dish were
already in the works by the time Tarkin first set eyes on it following
his promotion to the rank of Moff—escorted to Geonosis in utmost
secrecy by the Emperor himself.
Tarkin
Both sources agree that:
- Tarkin was perhaps “responsible” in the sense that he had brought the idea of a Death-Star type weapon before the rise of the Empire.
- Tarkin was a major force behind the “mobile battle station.”
- Tarkin was eventually to assume full leadership of the project.
Catalyst takes place before Tarkin. By the end of Catalyst, Krennic has been taken down a notch, and Tarkin has been granted significantly more authority over the project (the position we find him in in Tarkin):
Krennic snorted. “Is that why Tarkin has been assigned command of the
Sentinel bases?”
Amedda spread his large hands. “He is merely there to safeguard
against further setbacks. It pleases the Emperor to keep him close to
you.”
“That’s Tarkin’s job—to monitor me?”
Catalyst: A Rogue One Novel
The reason there’s no mention of Krennic or Erso is that there’s not much reason for them to show up. Krennic’s been embarrassed and relegated to a less prestigious position, and isn’t really causing any trouble. Erso’s just a scientist, albeit a high-profile one, and Tarkin probably doesn’t think about him very often.
