How did "stuck-up" get to mean "snob"?

According to Etymonline expressions using the concept of holding the nose up in the air suggesting superiority or disdain are used from 1570. Probably other expressions like stick one's nose up in the air and stuck-up are derived from this usage:

Nose:

• To turn up one's nose "show disdain" is from 1818 (earlier hold up one's nose, 1570s); similar notion in look down one's nose (1921).

Stick one's nose up in the air:

• Fig. to behave in a haughty manner. Jeff stuck his nose up in the air and walked out. Don't stick your nose up in the air. Come down to earth with the rest of us.

Source: McGraw-Hill Dictionary of American Idioms and Phrasal Verbs.

Ngram AmE, Ngram BrE show similar usage of 'stuck-up' since the beginning of the 19th century both in US and UK.

From: the Phrase Finder:

• 'Stuck-up' had emerged a century or so earlier (thanToffee-nosed ), and is found in Charles Dickens' Nicholas Nickleby, 1839:

• 'He's a nasty stuck-up monkey, that's what I consider him,' said Mrs. Squeers.

From The Edinburgh Review or critical journal for october 1829....january 1830

• They are comparatively, and without disparagement of their vast and almost superhuman merit, stuck- up gods and goddesses.

There does not seem to be a definite answer here, but it seems that most people are generally satisfied that stuck up may have come from the idea of sticking one's nose up.

However, I hope I can provide support for the idea that the phrase stuck up could easily have come from a meaning that has nothing to do with a person's nose.

Possibly stuck up came from ticket / etiquette

Etiquette is simply (and literally) 'a ticket' (in fact the English ticket, which originally meant any note, memorandum, voucher, etc., comes from this source). The word is derived from the French estiquer, meaning 'to stick'. The first rules prescribed by authority to be used in social or official life were 'stuck' up on the walls for all to see and follow. Word Origins

ticket (n.) 1520s, "short note or document," from a shortened form of Middle French etiquet "label, note," from Old French estiquette "a little note" (late 14c.), especially one affixed to a gate or wall as a public notice, literally "something stuck (up or on)," from estiquer "to affix, stick on, attach," from Frankish *stikkan, cognate with Old English stician "to pierce," from Proto-Germanic *stikken "to be stuck," stative form from PIE *steig- "to stick; pointed" (see stick (v.)). Online Etymology Dictionary

I'd like to emphasize the that etiquette meaning ticket comes from estiquer meaning to stick, because at one time all tickets were stuck up on walls for the community to read and, therefore, a ticket and something stuck up on a wall meant much the same thing.

So back when official rules and social rules were stuck up for all to read, the term stuck up could very reasonably have evolved to describe anything very superior and/or haughty, above the level of normal people. A person could be stuck up if that person were to behave as if they had, or actually had, official or social superiority.

Shakespeare did not specifically use stuck up to imply "conceited" to my knowledge but he did use some things that meant basically the same thing. For instance, in Much Ado About Nothing Act I Scene I (1598) the first words from Beatrice are:

"I pray you, is Signior Mountanto returned from the wars or no?" MAAN Act 1 Sc 1

where "montanto" is "a strike or jab made in an upward direction" Collins English Dictionary - a "sticking" in an upward direction, and the meaning of calling Benedict Mountanto is to imply that Benedict is conceited, as confirmed when Beatrice in her next words says:

"He set up his bills here in Messina and challenged Cupid at the flight." MAAN Act 1 Sc 1

Set up his bills means he stuck up tickets:

When a fencing-master visited a town he posted up bills setting forth his accomplishments, and the reasons why the world should learn fencing from him alone. Probably, too, these notices contained challenges to all who might feel inclined to have a bout with him. shakespeare-online.com

It is clear that Beatrice is insulting Benedict with these descriptions of him being, what we today would call stuck up, because Leonato replies:

"Faith, niece, you tax Signior Benedick too much" and soon after that Leonato explains "There is a kind of merry war betwixt Signior Benedick and her." MAAN Act 1 Sc 1

Similar to Shakespeare's language, Christopher Marlow's Dr. Faustus (1616) begins by describing the incredible success that Faustus has achieved and states:

"Are not thy bills hung up as monuments"? DrF Act 1 Sc 1 Ln 19

In 1878, the poem/song Sequel to Grandfather's Clock referred to the modern clock that replaced writer Henry Clay Work's grandfather's original closk as "vain, stuck-up thing on the wall" using stuck up with the double meaning of being vain and being up on the wall. This song was written near the end of Work's live (1832-1884) so as a boy he would have heard the phrase stuck up used very close to the time of it's first recording, and how he uses it may support the idea that stuck up came from a ticket/etiquette literally stuck up on a wall.

In 1909, in the book The Pilgrim's March by Sir Henry Howarth Bashford describes a situation where something being stuck up on a wall is a sign of accomplishment, and is connected to likely causing conceit in the person:

"'I came across some of that fellow's verses only this afternoon - stuck up on a cottage wall in the farthest corner of my parish'.....'I haven't told him about it,' he added presently, 'because he's conceited enough already.'"

Compare the Austrailian phrase "he has tickets on himself" which means the same thing as stuck up. The Oxford Dictionary of Moden Slang records this meaning of to have tickets on as "to have a high opinion" and especially to have tickets on oneself as "to be conceited" from 1908.

I believe etiquette / ticket is a more likely root of the term stuck up than one's nose, but I can't say it definitely is.

My Oxford English Dictionary gives this definition, dating it to the 19th century (UK): stuck-up /stʌkˈʌp/ adjective. colloq. E19 (= early nineteenth century = 1800-1829) [ORIGIN from stuck adjective + up adverb².]

Affectedly superior, pretentious, snobbish.

(Example) D. Madden: Stuck-up baggage…You're better off without her for a friend.

It doesn't really help you with the "nose in the air" idea (although I agree with you that that is probably the derivation) but it does give a date.

The first known recorded use of stuck-up is in 1829 (References: 1, 2). No one has definitive documentation on how its use started.

The idea that it involves having the nose stuck up in the air is very likely. This is a word-of-mouth explanation that has come down through the years. Before we can accept such a word-of-mouth explanation, we need some additional evidence. Couple the word-of-mouth with other phrases such as, "to look down ones nose at" and you have evidence that the nose is in other phrases that are used for the same connotation. This lends credibility to the theory. While this evidence is still weak, it is the best we have and it at least provides some corroboration to the word-of-mouth explanation.

Hence, until further evidence is presented (usually older written text), your belief is as correct as far as the little evidence there is suggests.

The term stuck-up is short for somebody's nose stuck up in the air.

It is completely possible that someone might investigating other theories or source phrases and prove this wrong someday.

Note: It would be nice to read the actual first use from 1829. I haven't found the text.