A blend of the adjective mad and Manchester, the name of an industrial city and metropolitan district in north-western England, the British-English word Madchester:
– used as an adjective: refers to the clubs, music, rock bands, etc., that proliferated in Manchester in the late 1980s and early 1990s;
– used as a noun: denotes Manchester as a centre of popular music and club subculture in Britain in the late 1980s and early 1990s; also denotes the music characteristic of Manchester at that time.

The earliest occurrences of the word Madchester that I have found are as follows, in chronological order:

1-: From Madchester Rave On (November 1989), the title of the second EP by the British rock band Happy Mondays. (This rock band was formed in Salford, an industrial city and metropolitan district near Manchester.)

2-: From Bunny-hopping into the Christmas charts, published in the Tamworth Herald (Tamworth, Staffordshire, England) of Friday 8^th December 1989:

A TAMWORTH company hope to be top of the pops this Christmas thanks to their inflatable bunnies, or to be precise, Jive-bunnies! Info Publishing, of Ladybank, Tamworth, have secured a deal to provide the people behind chart-topping act, Jivebunny ¹, with 1,000 inflatable bunnies to accompany the bands’ [sic] new single, widely tipped to be the biggest selling record this Christmas.
[…]
The blow-up bunnies are to be used as part of the promotional campaign by the record label Music Factory, for their Jivebunny in-store displays.
Info Publishing are already high in the charts with the indie pop Manchester band, The Happy Mondays. They were given the job of making an 11-feet long inflatable store display for Factory Records, to promote the band’s recent “Madchester” single.

¹ This refers to Jive Bunny and the Mastermixers, a pop-music act formed in 1989.

Photograph illustrating Bunny-hopping into the Christmas charts, published in the Tamworth Herald (Tamworth, Staffordshire, England) of Friday 8^th December 1989:

Chart toppers… (Left to right) Phil Moylan, Ian Buckley and Ian Gibbons, with the “Madchester” inflatable.

3-: From House-proud, by Len Brown, reporting from Manchester, published in the Observer (London, England) of Sunday 17^th December 1989:

Nowadays, it’s called ‘Madchester’, where hits are born, where the clubs are packed, and where the techno-beat of house hysteria is as incessant as the rain. If seeing is believing, then Manchester is where it’s happening.

4-: From Flare enough, by Andrew Martin, published in The Guardian (London and Manchester, England) of Monday 15^th January 1990:

There is a sentence that every fashion journalist must write at least once in his or her career. It is this: Flares are back.
[…]
[…] The real culprits are living in Manchester.
Manchester—or “Madchester”, as it is known to the hip—is the centre of a “scene”. At its core are two loutish pop groups, The Stone Roses ² and Happy Mondays. Musically, these groups represent the cross-fertilisation of the shaggy, guitar-jangling “indie” scene and the stylish minimalism of House music. But the important thing about these bands is that they wear flares, and so do their followers, producing a situation in which real live human beings have actually been seen walking up and down high streets wearing flared trousers.

² The Stone Roses were an English rock band formed in Manchester in 1983.

5-: From Mirror Live Special: MADchester, by Marcia Brackett, published in the Daily Mirror (London, England) of Tuesday 30^th January 1990:

The whole world is raving at a new hot spot
IT’S THE hottest place in Britain. It’s the city where it’s all happening. It’s MANCHESTER.
Every weekend coachloads of kids from the South spend Friday nights speeding North on the motorway, heading for what is now the trendiest spot in Britain.
And to emphasise the new flavour and appeal of this city once famous for Coronation Street ³, soccer and black pudding, they’ve renamed it MADchester.
[…]
ROSES GROW ON YOU
In their elephant flares and lager-lout shirts, bands like The Stone Roses—currently sweeping all the music polls as Best New Band—the Happy Mondays and The Inspiral Carpets ⁴ answer the teenage dream to be different.
As the MADchester scene gathers momentum, new groups everywhere are eager to board the band-wagon.
[…]
So trendy to go SCALLIDELIC!
The fashion look is called Scally. It used to be the term for naff smart but the Madchester kids have taken it and made it their own.
Now they go scallidelic by deliberately dressing naff with the emphasis on FLARES.

³ Coronation Street is a television soap opera created in 1960, set in Weatherfield, a fictional town based on Salford, near Manchester—cf. a Lancashire phrase: ‘the full monty’ and meaning and origin of the British phrase ‘big girl’s blouse’.
⁴ The Inspiral Carpets were a British rock band formed in 1983 in Oldham, an industrial town near Manchester.