The name Lushington has been used in various jocular phrases referring to alcohol consumption.

This name punningly alludes to lush, which:
– as a noun, denotes alcoholic drink;
– as a verb, means to consume alcohol.

James Hardy Vaux (1782-1841?), an Englishman who was convicted and transported to Australia on three separate occasions, gave the following explanations in A Vocabulary of the Flash Language, a glossary appended to Memoirs of James Hardy Vaux (London: Printed by W. Clowes, 1819):

LUSH, to drink; speaking of a person who is drunk, they say, Alderman Lushington is concerned, or, he has been voting for the Alderman.
LUSH, beer or liquor of any kind.

Lushington is jocularly used as a surname in the following, about the proprietor of a tavern, from Life in London; or, The Day and Night Scenes of Jerry Hawthorn, Esq. and his elegant Friend Corinthian Tom, accompanied by Bob Logic, the Oxonian, in their Rambles and Sprees through the Metropolis (London: Sherwood, Neely, and Jones, 1821), by the British journalist and author Pierce Egan (1772-1849)—heavy wet is slang for malt liquor:

No one could read his customers better than Mr. Mace. The attention he displayed towards any of his party, when Mr. Lushington had got the “best of them,” showed his judgement;—he had a butt of heavy wet prepared for the occasion, and also a cask of liquor, which gave considerable proofs of his kindness, that his articles should not be too strong for their already-damaged heads.

According to A New English Dictionary on Historical Principles; founded mainly on the materials collected by the Philological Society (Oxford: The Clarendon Press, 1908)—i.e., the Oxford English Dictionary:

The ‘City of Lushington’ was the name of a convivial society (consisting chiefly of actors) which met at the Harp Tavern, Russell Street, until about 1895. It had a ‘Lord Mayor’ and four ‘aldermen’, presiding over ‘wards’ called Juniper, Poverty, Lunacy, and Suicide. On the admission of a new member, the ‘Lord Mayor’ (of late years at least) harangued him on the evils of excess in drink. The ‘City’ claimed to have existed for 150 years […]. Our information is from ‘Sir’ B. Davies, the last ‘Lord Mayor of Lushington’.

The information given by ‘Sir’ B. Davies is partly confirmed by the following from Streetology of London; or, The Metropolitan Papers of the Itinerant Club. Being a Graphic Description of Extraordinary Individuals who exercise Professions or Callings in the Streets of the Great Metropolis; containing a True Picture of their Characters, Occupations, Haunts, and Adventures. Edited by Jack Rag, Esq., K.C.S.S. (London: James S. Hodson, 1837):

We shall give Dick Tynt’s description of the Temple-bar coach stand.
“[…] I had been to Common Garden playhouse,” says Dick, “to see Blue Beard cut off the heads of his wives […]. Well, after the funniment was over, away I scampered into the street again, and meeting with a few old palls, we found our way to the Harp, where beggars, ballad-singers, barbers, broken down lawyers, mechanics, and play-actors mingle together among goes of gin, heavy wet, and tobacco smoke. The regular frequenters of this notorious theatrical house, soon found out we were strangers in this land of the learned, and before we could swallow a pot of gattar [= porter], and puff out our charged yard of clay, we were invited by a red-face, pimple-nosed gentleman, to become members of the city of Lushington, and not minding a bit of a spree, for the sake of being let into the secrets of this city, and its constituency, especially as they boasted of Universal Sufferage, Vote by Ballot, and Radical Reform, we agreed to be made members at once. I had no sooner said the word, than the bells, for there is one at each table, were all rung in the most convulsive manner, and in came the waiter.
‘William,’ said the red-face, pimple-nosed gent., puffing a huge cloud of tobacco smoke from his pipe into his face, so that a pair of bellows were necessary, if we had desired to see his face, ‘Bring in four pots of porter for each table—there are four new members for the borough o’ Lushington.’
In a twinkling the gattar was placed before the motley members of the room, and William remained standing before the table whereat we were seated; not being awake to the business, I took no notice, when a little fellow, who feasted his nose with Scotch rappee every five minutes, and who sat next to me, dressed in a soldier’s hat and coat that had been dismantled of their trimmings, said, in an Irish brogue, which was rendered most unmusical, by being forced through his olfactory organs,—‘Fork out, my pinks, and pay for the lush, for upon my conscience, you are just now, for the first time in your life, honoured with the company of gentlemen, men of talent, upon my honour, sir, with whom the immortal Garrick would have been proud to have associated—yes, Sir, believe me, you are now with the great dramatic lights of the age. This is hallowed and classic ground, whereon you are. The Keans, the Kembles, the Cookes, the Bradleys, Blanchards, and Grimaldis, keep their courts here, and Momus and Bacchus preside at our festivals; therefore, pay my boy, pay, and let the waiter abscond, that we may proceed to business, and initiate you and your friends into the honourable franchises of the ancient city of Lushington.’
Thinking it useless to stand upon trifles, I tipped the rag for the heavy, which was no sooner done than about twenty voices commenced singing in the most vociferous manner:
“And we’ll chase the buffalo,
 And we’ll chase the buffalo,
 Through the wild woods we’ll wander,
 And we’ll chase the buffalo.”
“At the conclusion of this introductory stanza, Mr. Simes, or Simpkins, for I couldn’t catch his name exactly, rose, and made the following speech:—
“‘Gentlemen, electors of the ancient and honourable city of Lushington, it becomes my pleasing duty to inform you, that there are four most highly honourable and independent gentlemen, proposed to me by the deputies of the four wards of this city, viz. Poverty Ward, Juniper Ward, Lunatic Ward, and Suicide Ward, who are desirous of becoming members of the said wards, and I, therefore, have the gratification of recommending them to your enlightened and liberal notice, as men of principle, and every way qualified to maintain the dignity and respectability of our city, having in the most liberal manner paid the usual fee, and are now anxiously waiting to order in the customary quart of juniper essence to be sworn in.’
‘Such generosity,’ cried a squeaking voice in the corner, can only be repaid by an immediate admission into the city of Lushington, therefore I second the motion.’
The juniper was ordered, and I forked out and made it all right. This act being quite satisfactory to the knights of the battle-axe and blue flame, I was immediately escorted from my seat by two officers, armed with authoritative staves, to the head magistrate of the city, and in conjunction with my three friends, Fred Tipple the Shrimp man, Harry Gamble the Pieman, and Tim Sullivan the Muffin Merchant, were sworn in members of the four wards, in the following words, which we repeated after Mr. Tom Hollingsworth, the clerk of the city.
“We swear, so help us bread and cheese,
 Beef steaks, and such like grubbery,
 To drink strong beer instead of small,
 And wink at public robbery.
 We swear to keep the city’s laws,
 And help a brother Lushington;
 And if he wants a drop of gin,
 To tip the browns to get him one,
 For universal sufferage,
 And vote by ballot through the land,
 By Shakspeare and by Juniper!
 Most solemnly we swear to stand.
 We also swear to love the girls,
 To smoke and drink till life is done,
 To honour and obey the king,
 Like loyal sons of Lushington.
 We kiss the book and drink the gin,
 This solemn oath to ratify;
 For Lushington we’ll always vote,
 The Lushingtons to gratify.
“After this constitutional oath, which was certainly administered with legislative gravity, we were duly registered in the ward books, and the quart of juniper having disappeared down the throats of the honourable constituency, we were called upon to make our maiden speeches, which, after much apologizing, we declined, and during the general uproar of singing, reciting, whistling, tumbling, and dancing, bid adieu to
“These mouthy kings of Thespian’s art,
 Who nightly meet to play their part;
 Making their lives a comic play,
 And mimicking the world away.””

The earliest occurrence of the phrase the City of Lushington that I have found is from the following illustration by the British artist Theodore Lane (1800-1828), published in The Life of an Actor (London: Printed for C. S. Arnold, 1825), by the above-mentioned British author Pierce Egan:
—In this illustration, a sign above the door reads The City of Lushington; two notices are on the wall: one, on the left-hand side, reads Suicide Ward, the other, on the right-hand side, reads Poverty Ward. This illustration is captioned “House of Calls for Actors.—Proteus visits the Harp; receives a nod from the Agent; and also witnesses a “fresh caught Hero” exhibiting his talents for a Country Engagement.”:

This illustration refers to the beginning of Chapter II, “Proteus pays a Visit to the House of Call for Actors, in company with Mr. Horatio Quill.”, which mentions the Harp Tavern:

Proteus, under the guidance of his friend Horatio Quill, visited the House of Call for Actors […].
Fortunately for Proteus the first visit he paid to the above resort of performers was during Passion Week. It is a short period of relaxation for the profession; and the house generally overflows during the week with talents of every description. […] Country managers, by dozens, may be witnessed on the “look out” for recruits; and itinerant actors, equally anxious and on the alert to procure new engagements, likewise repair to the metropolis, and direct their attention towards the Harp.

There are several other mentions of the Harp Tavern in The Life of an Actor: for example, “Another Visit to the Harp: a new Engagement the Result.” (Chapter IV), and “Proteus appears at Richardson’s Theatre, in Bartholomew Fair. One Pill a Dose. Looks in at the Harp after Business.” (Chapter V).

The second-earliest occurrence of the phrase the City of Lushington that I have found is from the account of a court case, published in The True Sun (London, England) of Friday 28^th December 1832:

An old woman who has been for several years receiving parish relief and parish clothing was brought up, charged with having got drunk and abused the parish authorities.
The Lord Mayor—l am sorry to hear such an account of an old parishioner who has been so well treated.
The Beadle—My Lord, she has by her conduct lost 2l. worth of Christmas clothing which the churchwardens were about to send her.
The Old Woman—Please you, my Lord, if the poor had their rights, the churchwardens and overseers would not stuff themselves and get drunk so often together.—(Laughter.)
The Lord Mayor—l hope they do no such thing, and I believe you say that to excuse yourself.
The Old Woman—No, my Lord. They belongs to the reg’lar city of Lushington. They are fond of the lush, and if they finds any of us overcome with hunger and a little drop of Jackey, they ’mediately begins for to talk morals to us, and they refuse to tog us for the winter.—(Loud laughter.)
The Lord Mayor said that if he received a solemn promise of amendment from the aged pauper he would interfere with the parish authorities in her behalf.
The Old Woman—God bless your Worship; and I think we’d better not say a word about the lush.—(Laughter.)