In the book, it's because he'd only just arrived, on horseback, evidently having accompanied Grond (and some additional troops) to the battlefield.
Over the hills of slain a hideous shape appeared: a horseman, tall,
hooded, cloaked in black. Slowly, trampling the fallen, he rode forth,
heeding no longer any dart. He halted and held up a long pale sword.
And as he did so a great fear fell on all, defender and foe alike; and
the hands of men drooped to their sides, and no bow sang. For a moment
all was still. The drums rolled and rattled. With a vast rush Grond
was hurled forward by huge hands. It reached the Gate. It swung. A
deep boom rumbled through the City like thunder running in the clouds.
... And suddenly upon the last stroke the Gate of Gondor broke. As if
stricken by some blasting spell it burst asunder: there was a flash of
searing lightning, and the doors tumbled in riven fragments to the
ground. In rode the Lord of the Nazgûl. A great black shape against
the fires beyond he loomed up, grown to a vast menace of despair. In
rode the Lord of the Nazgûl, under the archway that no enemy ever yet
had passed, and all fled before his face.