Was it Gared?
The survivor is named differently in show and books.
- In books he is named Gared.
- In Show he is named Will
Other than that, the survivor is most definitely Gared.
The commander of the party, Ser Waymar Royce had ordered Gared to stay behind with the horses while he went ahead with Will.
From AGOT, Prologue:
“If I need instruction, I will ask for it,” the young lord said.
“Gared, stay here. Guard the horses.”
Ser Waymar himself got killed in a fight with the others, Will observed his corpse:
Royce’s body lay face down in the snow, one arm outflung. The thick
sable cloak had been slashed in a dozen places. Lying dead like that,
you saw how young he was. A boy.
Then Will saw Royce rise up again and throttle him to plausibly death:
Will rose. Ser Waymar Royce stood over him.
His fine clothes were a tatter, his face a ruin. A shard from his
sword transfixed the blind white pupil of his left eye.
The right eye was open. The pupil burned blue. It saw.
The broken sword fell from nerveless fingers. Will closed his eyes to
pray. Long, elegant hands brushed his cheek, then tightened around
his throat. They were gloved in the finest moleskin and sticky with
blood, yet the touch was icy cold.
So it is clear from here that out of the three members of the party, two got killed in the prologue. Which leaves us with Gared.
In AGOT, Tyrion III, Lord Commander confirms that it was Gared who got executed by Eddard Stark:
The Lord Commander took no notice of the irritating bird. "Gared was
near as old as I am and longer on the Wall," he went on, "yet it would
seem he forswore himself and fled. I should never have believed it,
not of him, but Lord Eddard sent me his head from Winterfell. Of
Royce, there is no word. One deserter and two men lost, and now Ben
Stark too has gone missing." He sighed deeply.
Even Craster knew that Gared had been executed. From ACOK, Jon III:
Lord Mormont said, "Ben was searching for Ser Waymar Royce, who'd
vanished with Gared and young Will."
"Aye, those three I recall. The lordling no older than one of these
pups. Too proud to sleep under my roof, him in his sable cloak and
black steel. My wives give him big cow eyes all the same." He turned
his squint on the nearest of the women. "Gared says they were chasing
raiders. I told him, with a commander that green, best not catch 'em.
Gared wasn't half-bad, for a crow. Had less ears than me, that one.
The 'bite took 'em, same as mine." Craster laughed. "Now I hear he
got no head neither. The 'bite do that too?"
But how did he got across the wall?
See my previous answer here which discusses all the ways wall can be defeated in detail.
Being veteran member of Night's Watch, he (Will in Show and Gared in books) should have been able to use the magical black gate in Night Fort however it is not known if he knew about the gate. Alternatively he could have just used any of the methods listed in my other answer, depending on tools he had with him and the location he was at. It is not specifically mentioned.