In Peaky Blinders (set in 1919), the actress who plays Grace had a pretty obvious nose job between the first two seasons (1919-1921, I think?). Would this kind of cosmetic plastic surgery be a thing at that time? Would it have been that successful, or would it have left severe scarring?

The cultural and intellectual historian Sander Gilman has a book called Making the Body Beautiful that explicitly addresses the history of cosmetic surgery in the twentieth century (and before). In it, he relates the history of rhinoplasty and other nasal surgery quite extensively. Nasal reconstructive surgery was actually one of the earlier aesthetic surgeries available, as one of the unfortunate symptoms of untreated syphilis is the deterioration of the nose (you can Google it if you like - it's pretty gruesome). Since at least the 17th century, people have been trying to repair this deformity, at first usually through the insertion of a replacement nasal bridge made of gold or some other durable material. Another thing they tried was the grafting of a flap of forehead or other skin over the damaged nose in order to try to hide it. Keep in mind that all these surgeries happened without modern sanitation and anesthesia. These surgeries got more and more elaborate and sophisticated, especially as the nineteenth century turned into the twentieth, and it became important not just to repair the nose, but to repair it in such a way that an observer could not tell that it was a specifically syphilitic nose that had been repaired.

Of course, what you're asking about isn't really reconstruction of a diseased nose: it's a purely aesthetic attempt to re-shape the nose. Given the nineteenth century European obsession with race and racial hierarchy, this was also a going concern. Being able to "pass" as a "superior" race was a big deal - for an Irishman to pass as Anglo, for example, could spur his elevation into a higher social and economic class. By the turn of the twentieth century, medical technology was sophisticated enough to render these procedures relatively safe, and they became more common. John Orlando Roe, a surgeon from Rochester, NY, became well known for his operation to cure what at the time was called the "pug nose," for example.

So in essence, yes, plastic surgery was available during the early twentieth century to "correct" perceived nasal defects. One final note: Gilman says that even as the procedures became safer and more able to replicate "natural" noses, the visibility of a scar was still a source of stigma. In essence, evidence that one had received plastic surgery was as undesirable back then as it is today.