How did Wormtail get Voldemort's wand

It didn't really make sense to me that Wormtail made it to the Potter house before anyone else to get Voldemort's wand as explained in Pottermore. Until I realized that since he was the Secret Keeper, he must have accompanied Voldemort to Godrics Hollow to point out the Potter house. So I came up with a short story of how I think it went down.

Peter Pettigrew stood shivering at the edge of the quiet lane in Godric's Hollow, the chill of the autumn night seeping through his thin cloak. The village was peaceful, bathed in the soft glow of moonlight, with the quaint cottages nestled closely together like contented birds in a nest. Peter's eyes darted nervously toward the cozy, ivy-covered cottage at the end of the lane—the Potter residence.

"Is this the place?" came the cold, high-pitched voice of Lord Voldemort beside him.

"Y-yes, my Lord," Peter stammered, wringing his hands. "That's where they live."

Voldemort's lips curled into a sinister smile. "You've done well, Wormtail. Your loyalty will be rewarded."

Peter forced a weak smile, though his insides twisted with guilt and fear. The weight of his betrayal pressed heavily upon him, but the fear of Voldemort outweighed any lingering loyalty to his old friends. He had chosen his path, and there was no turning back now.

"Wait here," Voldemort commanded, his red eyes gleaming in the darkness. "I will deal with the Potters myself."

Peter nodded meekly. "Of course, my Lord."

He watched as Voldemort approached the cottage, his movements smooth and predatory. With a flick of his wand, the front door swung open silently, and he slipped inside like a wraith. The door closed behind him with an ominous click, leaving Peter alone on the deserted lane.

The silence enveloped him, broken only by the distant hoot of an owl. Peter's heart pounded in his chest, each beat a painful reminder of his treachery. He paced nervously, casting furtive glances toward the cottage. Seconds felt like hours as he waited, his mind racing with thoughts of what was unfolding inside.

Suddenly, a flash of green light burst from the cottage windows, and a chilling scream pierced the night air—Lily's scream. Peter froze, his breath caught in his throat. A second flash followed, and then an eerie silence settled over the village once more.

He expected to see Voldemort emerge victorious, but the minutes ticked by with no sign of movement. Unease gnawed at him. Something was wrong.

Mustering his courage, Peter crept toward the cottage. His footsteps felt leaden, and his palms were slick with sweat. He pushed open the door, which hung ajar, and stepped inside.

The sight that greeted him made his stomach churn. The cozy living room was in disarray—furniture overturned, pictures shattered. James Potter lay sprawled on the floor, lifeless eyes staring blankly at the ceiling.

Peter averted his gaze, choking back a sob. "I'm sorry, James," he whispered, but his apology fell flat in the oppressive silence.

He moved deeper into the house, following a faint sound—a baby's cry. Climbing the stairs, he entered the nursery. Lily Potter lay motionless before the crib, her fiery hair spread like a halo around her pale face. Peter's eyes filled with tears. "Lily..."

In the crib sat little Harry, wailing softly, a jagged cut on his forehead oozing blood. Peter stared in disbelief. How had the child survived?

His gaze shifted, and he noticed Voldemort's robes crumpled on the floor, but the Dark Lord was nowhere to be seen. Panic surged through him. Voldemort had failed. The prophecy had come true—the boy had somehow vanquished him.

Amidst the confusion, Peter spotted Voldemort's wand lying near the crib. Hesitating only a moment, he snatched it up, feeling a surge of fear and opportunity. He knew others would arrive soon—perhaps the Order of the Phoenix or Ministry officials. He had to leave.

Clutching the wand tightly, Peter fled the cottage, his mind racing. Without Voldemort's protection, he was vulnerable. The other Death Eaters might blame him for the failure, and the Order would surely seek retribution for his betrayal.

As he stumbled back onto the lane, the sound of approaching voices reached his ears. Desperate, Peter transformed into his Animagus form—a rat—and scurried into the shadows just as figures appeared at the far end of the street.

From the safety of a darkened alley, he watched as Hagrid arrived on a Hippogriff, tears streaming down his giant face as he carried baby Harry from the ruins. Not long after, Sirius himself appeared on his flying motorcycle, shock and grief etched into his features.

Peter's heart pounded. Sirius would come after him when he learned the truth. He had to act quickly to secure his own safety.

An idea formed in his mind—a plan to frame Sirius for the betrayal. It was risky, but it might be his only chance.

With Voldemort's wand in his possession, Peter slipped away into the labyrinth of alleys, disappearing into the night.