Born in Edinburgh, Scotland, Welsh started playing in the teenage Leith Silver Band and gigged with Archie Semple's Capital Jazz Band. After moving to London in the early 1950s, Welsh formed his own band. The group, with clarinetist Archie Semple, pianist Fred Hunt, trombonist Roy Crimmins and drummer Lenny Hastings, was a backbone of British traditional jazz in the 1950s, playing a version of "Chicago style" jazz. In the 1960s, Welsh's band invited guest stars such as Earl Hines, Red Allen, Peanuts Hucko, Pee Wee Russell and Ruby Braff to perform with them. Released on 10 August 1961, the Alex Welsh Band's track "Tansy", peaked at #45 in the UK Singles Chart. In January 1963, the British music magazine NME reported that the biggest trad jazz event to be staged in Britain had taken place at Alexandra Palace. The event included George Melly, Diz Disley, Acker Bilk, Chris Barber, Kenny Ball, Ken Colyer, Monty Sunshine, Bob Wallis, Bruce Turner, Mick Mulligan and Welsh.
Welsh toured internationally and played at the 1967 Antibes jazz festival, the 1968 Newport Jazz Festival and 1978 Nice Jazz Festival. He was often considered as not just a trumpeter but also a good entertainer often adding a little humour and rapport with his big band sound. He performed regularly in the 1970s at the Bell Public House in Maidenhead, Berkshire.
Albums such as Music of the Mauve Decade, Strike One, and Echos of Chicago have been re-issued on CD.
- Dixieland to Duke/The Melrose Folio (1957/60)
- Music of the Mauve Decade (1960)
- Classic Concert, (1971)
with Peanuts Hucko
- Peanuts Hucko Vol.1 (Lake Records, 2002) recorded in 1967