Born in New Zealand and trained as a journalist, Currie emigrated to the UK in 1977. Currie squatted in various places in South West London, ending up in Lillieshall Road, Clapham Old Town. In 1979, with her across-the-road neighbour, Trace Newton-Ingham (Traci Newton), she co-founded the dread-punk-improvising group, The Unfuckables. The Unfuckables performed one "gig" at an anti-psychiatry conference in early 1980, held in the Conway Hall, Red Lion Square, London. For this particular gig the group comprised the two co-founders, plus an array of musicians from London's "underground" music scene — Viv Albertine (The Slits), Gareth Sager (The Pop Group), Charles Bullen and Charles Hayward (This Heat), Tom Bailey (of Thompson Twins, whom she would later marry), Jimmy Cauty (later of The KLF, and her current husband) and improviser Steve Beresford, amongst others.
In 1981, Currie joined Tom Bailey, Joe Leeway, and others to form part of the Thompson Twins, the line-up of which included up to seven members in its early days. The Thompson Twins became a trio in 1982 and signed two major record contracts with Arista Records before signing with Warner Bros. Records. Currie was a lyricist, percussionist, visual stylist and singer in the band and co-wrote and recorded 6 albums which included gold and platinum records and the hits Doctor! Doctor!, Hold Me Now, and You Take Me Up. The band performed at the JFK Stadium, Philadelphia for the 1985 Live Aid concert and worked with artists including Nile Rodgers, Madonna, Grace Jones, Alex Sadkin and Jerry Harrison of the Talking Heads amongst others.
In 1984 the band participated in the "first international satellite installation" by Nam June Paik, Good Morning, Mr. Orwell.
Her songwriting credits also include "I Want That Man", an international hit for Deborah Harry in 1989.
By 1992, Currie and her husband, fellow Thompson Twins band member Tom Bailey, had grown tired of being expected to chase chart success with new Thompson Twins material. After touring India collecting inspiration and audio samples, they elected to form Babble, featuring Currie as lyricist, percussionist and visual artist, as a means of creating music without the commercial expectations that were placed on the Thompson Twins. By 1994 Babble had released their first album. Currie later returned to New Zealand working primarily as a glass artist and environmental activist. She was founder of the women's anti-genetic engineering movement Mothers Against Genetic Engineering in Food and the Environment (MAdGE). In 2004 she designed a series of protest billboards that caused controversy in New Zealand but won several international art / science awards.
In 2004 she returned to London where she now works under the name Miss Pokeno and continues to make protest art and build chairs that "tell dark tales". She is also a member of the Sisters of Perpetual Resistance, who make "militant feminist art", and the Armchair Destructivists.
Alannah was married to fellow band member Tom Bailey from 1991 to 2003. They have two children, Jackson (b. 1987) and Indie (b. 1992).
In 2011, Alannah married Jimmy Cauty (formerly of the KLF).