I'm a-gonna give you (nearly) all the answers in the world. Likely more than you wanted. This is an extremely clever & funny song in its origins but the references (as you've discovered) are now so obscure that not even the Dubliners get it right any more. You also need to have the earlier text - there are several Mondegreens that have snuck into most renditions that obscure the obscure more.

TAKE ME UP TO MONTO
(George Hodnett)

Well if you've got a wingo take me up to ringo
Where the waxies sing-o all the day
If you've had your fill of porter and you can't go any further
Then give your man the order back to the Quay.

Chorus:

And take her up to Monto, Monto, Monto,
Take her up to Monto, langeroo to you.

You've heard of butcher Foster, the dirty old imposter
He took a mot and lost her up the Furry Glen
He first put on his bowler, then he buttoned up his trousers
And he whistled for a growler and he said, "My men,

Chorus: Take me up to Monto, etc.

The fairy told him, `Skin the Goat,' O'Donnell put him on the boat
He wished he'd never been afloat, the dirty skite
It wasn't very sensible to tell on the Invincibles
They took aboard the principals, day and night.

Chorus: Be goin' up to Monto, etc.

You've seen the Dublin Fusiliers, the dirty old bamboozaliers
They went and got the childer, one, two, three
Marchin' from the Linen Hall, there's one for every cannonball
And Vicky's goin' to send youse all o'er the sea.

Chorus: But first go up to Monto, etc.

When the Czar of Rooshia, and the King of Prooshia
Landed in the Phoenix in a big balloon
They asked the Garda Band to play, `The Wearin' O' the Green'
But the buggers in the depot didn't know the tune.

Chorus: So they both went up to Monto, etc.

The Queen she came to call on us, she wanted to see all of us
I'm glad she didn't fall on us, she's eighteen stone
`Mr. Neill, Lord Mayor,' says she, `Is this all you've got to show to me?'
`Why no, ma'am, there's some more to see - pog mo thoin

Chorus:

And he took her up to Monto, Monto, Monto,
Took her up to Monto, langeroo - Liathroidi to you.
 

Monto (short for Montgomery St., near the customs-house) was, up until 1926,
a largish red-light district which featured some 1600 or so prostitutes.

I quote some research I found in Dublin:

Montgomery Street, near the Custom House, was reputed to be the biggest red-light district of its kind until its closing down occurred in 1925. The song itself, with its child-like, almost nursery-rhyme style delivery, is quite amusing but if the words are examined, it can be seen to be quite a clever and sometimes very sharp view of some recent historical events. The first verse is principally praising alcohol. In the second verse "Butcher Foster - the imposter", is Chief Secretary Forster, more usually known as "Buckshot". He had introduced Coercion Acts in the late 19th Century which allowed people to be arrested and imprisoned on suspicion of being involved in criminal activity. He was not a very popular individual which can be seen in the unfavourable way he is presented in the song. The bowler connects him to the crown and to loyalism, the growler to the English "Bulldog".

"Skin the Goat" was the nickname of James Fitzharris, the cabman who drove the murderers of Lord Cavendish and T.H.Burke to and from the Phoenix Park. He was sentenced to penal servitude for conspiracy because he refused to identify his passengers. Patrick O'Donnell, in another song was "a deadly foe to traitors". He had met the informer James Carey, who although he had played a leading in the murders, was freed for turning Queen's Evidence. Of the 27 members of the Invincible society who were arrested, Carey's evidence helped to send six for execution. Carey was then secretly dispatched to South Africa by sea and met O'Donnell "afloat". Then while travelling to Durban from Cape Town on the "Melrose" O'Donnell killed Carey and was sent back to London, tried and sentenced to death.

The Dublin Fusiliers come in for abuse also, and are mentioned in connection with the Boer War "oe'r the sea". The new police force, An Garda Siochana, come under suspicion too because their loyalty to the new "Gaelic" state is questioned when they can't play a nationalist melody. Queen Victoria comes in for the greatest abuse of all in the song when she is described unfavourably and is also grossly insulted in a most crude manner by the Lord Mayor of the city, before bringing her up to Monto!

per "THE WAXIES DARGLE," Waxies are candlemakers
per Dictionary of Slang, 'mot' equals any of girl/wench/doxy
per Brendan Behan, "mot" is Dublin slang for girl/woman (even mother) - perhaps from the Latin for mater

The in DT version agrees with Luke Kelly's.

In verse three, 'Garda Band,' not 'polismen' will be accurate but since they were police, it doesn't matter.  Line 4, however, should certainly be 'Vicky' and that's clearly what Kelly sings.  DT, we conclude, has misprinted.

Other differences may be attributed to Process but the explanation below does certainly support (or follow???) the printed source above.
 

Take Me Up to Monto
Words & music by George Hodnett