Perhaps the most common misconception about Voldemort is that he only got his snakelike red-eyed appearance after his resurrection. I get why people may assume this (an unfounded assumption typically based on the fact that snake venom was used, and that he was made in a cauldron, and the misunderstanding that Wormtail was an unskilled sorcerer), but there’s lots of evidence that he already looked like that in 1981, and nothing in canon implies his appearance was changed by the rebirthing potion.
In the graveyard, Voldemort calls gaining his current body getting his 'old body back’:
Note that the body creation potion is a pre-existing piece of old magic that Voldemort already knew about. He knew that to get his old body back, he would need to use this ancient potion with the three powerful ingredients. This is not magic he invented himself. Snake venom is not an ingredient in his rebirthing potion, but only an ingredient (along with unicorn blood) in the potion he was drinking while in the rudimentary body. He is not 'made of' snake venom any more than he's made of unicorn blood.
He tells a pretty thorough story of the last 13 years to the DEs, and never implies he looks different and explains why. If he did look different, he likely would have included that in his several pages of explanations of what had happened to him.
This happens right after he summons his DEs to the graveyard and they immediately recognize him—in the dark, from a distance so far away that they have to walk toward him and then crawl the rest of the way.
Fudge and other Ministry workers also immediately recognize him the following year. Harry did publish an interview describing him, but Fudge doesn't trust Harry at that point, or isn't claiming to anyway. The only thing that convinces Fudge is seeing a person he personally believes to be Voldemort with his own eyes. This is a person Fudge recognizes beyond doubt:
In Sorcerer's Stone, Voldemort's face is described:
He already has the white skin, the red eyes, and the slit nostrils years before the snake venom drinking and the rebirthing potion. They did not appear for the first time after he stepped out of the cauldron.
On Halloween 1981, we know two things about his appearance: that his hand is white, and that his face is so terrifying it makes a child run away.
We also know that James recognizes Voldemort immediately. I don't think his face itself would've been well-known, as there aren't any instances in the Second War of Voldemort being photographed, or images of him being published in the paper or hung up with the Azkaban escapees' portraits (only the verbal description in the Quibbler article). It seems more likely that what makes him so recognizable is that he has very unique features.
We know for a fact that by the late 1960s (ish?), Voldemort was no longer the handsome Tom Riddle. He was described like this at his interview with Dumbledore:
Notice the similarities to his appearance in SS & GoF: the white skin, the partially red eyes (the whites & flashing red multiple times), the slitlike nostrils. He seems to be partway through a transformation between the handsome human form of his youth and the snakelike form of his middle age. Harry says Voldemort's features here are not yet 'as' snakelike, which implies they are reminiscent of it, just not fully identical. I see no other possible explanation for Voldemort having somewhat snakelike features and slitlike nostrils and reddening eyes in ~1968, besides him slowly undergoing a visual transformation into the appearance we know. (Well, it could either happen very gradually or it could happen in distinct stages after each Horcrux creation, based on Dumbledore's info in HBP that the Horcruxes are what caused the appearance changes. Either way, it happened over many years.)
(A decade earlier in the Hepzibah Smith memory, Voldemort looks quite normal and like his younger self other than his eyes sometimes flashing red.)
There are around 13 years between this interview and Halloween 1981—years in which we must assume Voldemort's appearance continues changing along the lines it's already changing in 1968 and into the face we know from SS & GoF. Not only does 1968 Voldemort seem well on the way to his canon-era face, I just don't see another explanation for why he would have this face in 1992 if he did not in 1981.
What did Voldemort look like when he killed the Potters? Exactly like he did when he stepped out of the cauldron. Any claims otherwise are headcanons based on conjectures of the effects of the snake venom and the rebirthing process that are not supported by the text.
At no point in the First War was Voldemort handsome in the way he was in his youth. This means that anyone that Voldemort only met after his presumable return to Britain in the late 60s (such as perhaps Bellatrix) never knew him by his old face, only by some version of his new one.
