I don’t know if there’s also a word in English, but in German, there’s the term “Säufernase” (drunkard’s nose, sometimes also called “Schnapsnase”, I don’t think I have to translate that) for an unnaturally big and red nose some alcoholics develop.
I’m sure there is an English word for that, but it’s probably somewhat old-fashioned. For whatever reason, the trope of depicting a drunkard with a red nose has fallen out of favor (I imagine that the symptom itself is just as common as it always was; it’s just not as much in the public consciousness).
Not quite what you are looking for but if a former opioid addict needs to take opioids again due to significant trauma, post-op status, or malignant pain, it’s said they’re having a freelapse. I’ve had a few of these freelapses since I got clean and sober due to medical issues, and they don’t count against one’s recovery time. It is a risk to one’s recovery to go through such a thing, but with support from one’s clean and sober community one can avoid returning to active addiction.
You’re probably thinking of rhinophyma, characteristically associated with W.C. Fields.
It’s entirely possible to have rhinophyma (a complication of rosacea characterized by skin inflammation and enlargement of sebaceous glands affecting the nose) without being an alcoholic, though research cited in the linked article suggests that rhinophyma tends to be worse in heavy drinkers.
During Prohibition, the effects of adulterated Jamaica ginger led to its victims being known as having the “Jake walk” or “Jake walk blues”. Jamaica ginger extract in those days had high alcohol content and was popular in cocktails despite having a bitter taste. A major manufacturer used a chemical adulterant to make the flavor less unpleasant, thinking the additive was non-toxic. It turned out to be a neurotoxin. Tens of thousands experienced partial paralysis leading to a distiinctive abnormal gait, often permanent.
I found a modern recipe online for a “Jake Walk” cocktail using such things as tequila, rum and cane juice - but no Jamaica ginger.
Another famous sufferer was financier J. Pierpont Morgan. (An urban legend goes that a society matron repeatedly warned her family in advance of a visit by Morgan to not even say the word “nose” in his presence - and then later asked him while serving tea “Will you have one or two lumps in your nose?”)
There’s also the German word “Kater” (tomcat) for a hangover, and in my regional dialect “Affe” (monkey) means the same, while in rather urban, junkie circles it means heroin withdrawal symptoms.