In British English:
– the colloquial noun toffee-nose designates a snob or supercilious person;
– the colloquial adjective toffee-nosed means snobbish or supercilious.

The reason the noun toffee is used in those two words is unclear. This noun perhaps refers in fact to the informal British-English noun toff, which was originally used, in the mid-19^th century, among people of lower social status to designate a fashionable upper-class person. Perhaps, in toffee-nose and toffee-nosed, the image is of someone who, considering themself superior, keeps their nose high in contempt for the lower classes.

This may be supported by the alternative forms toffy-nose and toffy-nosed (toffy means characteristic of a toff). The earliest occurrence of toffy-nosed that I have found is from the following paragraph, from “Tommy and Jack.” Dealing also with the Discharged Sailor and Soldier and their Dependents, published in John Bull (London, England) of Saturday 7^th June 1919:

Toffy-Nosed Eastbourne:—Hospital patients at Eastbourne complain to us once again. The Nabobs of the place and their wives are against the men using the promenades, regarding the poor chaps as simply horrid. Even legless heroes in bath chairs are not permitted to get close enough to hear the band, as their chairs would obstruct the patriotic pedestrians. Eastbourne can do better than this.

The earliest occurrence of toffy-nose that I have found is from the account of a police-court case, published in the South Western Star (London, England) of Friday 3^rd March 1933:

P.C. Jones, 231L, said that at 10.30 on Thursday night he saw prisoners playing two instruments. They were under the influence of drink and were not playing properly. He told them they could not play there and they must go away. They argued and he told them if they did not go he would take them into custody. Gillman said, “You toffy nose, I’ll jam this trumpet under your nose.”

The form toffy-nosed also occurs, for example, in the following from the Belfast Telegraph (Belfast, Antrim, Northern Ireland) of Friday 31^st August 1956:

‘Toffy-nosed’ neighbours are really lonely

In East London the family still stuck together, and the neighbours were friendly, but out on the new housing estate in Essex there was no family to turn to, and the neighbours were “toffy-nosed,” Dr. Michael Young told the Psychology Section of the British Association at Sheffield to-day.
Dr. Young, who is Director of the Institute of Community Studies, Bethnal Green, was describing a three years’ survey in which a thousand people had been seen.
[…]
“Our informants spoke time and time again about the unfriendliness of the place. As one man said: ‘They’re all Londoners here, but they get toffy-nosed when they get here. They’re not so friendly.’”

The earliest occurrences of toffee-nose and toffee-nosed that I have found are as follows, in chronological order:

1-: From the caption to the following cartoon by Peter Fraser (1888-1950), published in Punch, or the London Charivari (London, England) of Wednesday 2^nd December 1914:

The Victor (after being admonished for un-scoutlike behaviour). “Well, you may say what you like, Sir, but I consider it distinctly subversive of discipline for an ordinary private to call his patrol-leader ‘Toffee-nose.’”

This caption was reprinted in several British newspapers in December 1914, which may have popularised the noun toffee-nose.

2-: From English Army Slang as used in the Great War, compiled from words sent in by various correspondents, published in Notes and Queries: A Medium of Intercommunication for Literary Men, General Readers, etc. (London: Published by The Times Publishing Company, Limited) of Saturday 10^th December 1921:

Toffee-nosed. Stuck up. (Trenches.)

3-: From the account of a police-court case, published in the South Western Star (London, England) of Friday 8^th June 1923—Arding & Hobbs was a department store in London:

Before Mr. Marshall at the South Western Police Court on Saturday Charles Arthur Day (40), navvy, no fixed abode, was charged with being drunk and disorderly at St. John’s Hill and with assaulting Sidney Mason, a tram conductor […].
[…]
The conductor said prisoner boarded his car outside Arding and Hobbs, saying “Is this a 34?” Prosecutor told him it was a 28. Prisoner, with foul language, then said “Who are you talking to?” On being requested to leave he said “Not for you, you toffee-nosed pig.”

4-: From a letter to the Editor, published in the Sheffield Daily Telegraph (Sheffield, Yorkshire, England) of Tuesday 12^th June 1923:

Sir,—[…] I personally have never booked a court on the Bolehills this season. I am under no delusions whatever about the clubs. As I said in my first letter the idea of inter-park play was a commendable one; but what I do object to is the “toffee-nosed” business. And there’s “bags of it” on the Bolehills.—Yours, etc.,
Alfred E. Ward.

5-: From Soldier and Sailor Words and Phrases (London: George Routledge and Sons, Ltd., 1925), by Edward Fraser and John Gibbons:

TOFFEE-NOSED: Stuck up.

6-: From a letter that the British intelligence officer and author Thomas Edward Lawrence (1888-1935) wrote to Herbert Baker on Friday 20^th January 1928—as published in The Collected Works of T. E. Lawrence (Lawrence of Arabia) (Musaicum Books, 2017):

I encouraged Graves, to give my reputation the coup de grace. A premature ‘life’ will do more to disgust the select and superior people (the R.A.F. call them the ‘toffee-nosed’) than anything.