Before becoming a professional musician, Fairweather was a librarian and has retained a strong interest in jazz bibliography and archiving.
He has been a full-time jazz musician since 1 January 1977, and Fairweather worked part-time for seven years before that with local jazz bands in Essex and London. He recorded his first album in 1975 with trumpeter Alex Welsh's band, as deputy for Welsh. After turning professional, Fairweather helped found pianist Keith Nichols' "Midnite Follies" Orchestra, also performing with Nichols in his "Ragtime Orchestra" and on a tribute show to Fats Waller. He performed with several other bands, including one led by drummer Lennie Hastings. Starting in 1978, he recorded solo albums, and joined a quartet known as Velvet, with guitarists Denny Wright and Ike Isaacs plus bassist Len Skeat.
Apart from his playing and group leading, Fairweather has long pursued a parallel career as a jazz broadcaster and writer. From 1985 to 1988, he worked and recorded with Brian Priestley's Special Septet and Tony Milliner's Mingus Music, and wrote the book How to Play Trumpet. By 1990 he had embarked on a dual vocation as broadcaster for BFBS, BBC World Service, Jazz FM (1991–92) and BBC Radios 2/3 (1992–98), including occasionally deputising for Humphrey Lyttelton on his show Best of Jazz and successively presenting Jazz Parade and Jazz Notes. In 1987 Fairweather founded the Association of British Jazz Musicians and the National Jazz Archive. That same year he formed the Jazz Superkings with Dave Shepherd, Brian Lemon, Allan Ganley and others. Fairweather helped bring jazz musicians into the British Musicians' Union by proposing, then founding, its Jazz Section in 1992 (discontinued in 2014). In 1994 (with trombonist Pete Strange), he co-founded The Great British Jazz Band and continues to teach and do solo work. Following the death of Humphrey Lyttelton, he was invited to succeed Lyttelton as the patron of the Birmingham International Jazz Festival. In 2013 he continues to lead his Half Dozen (winners of the Best Small Group category at the 2012 British Jazz Awards) and to perform as a soloist. In 2013 he received the Worshipful Company of Musicians Lifetime Achievement Award for Services to Jazz.