His father was of Polish descent. Dreja was born in Surbiton, and raised in Kingston upon Thames. His brother Stefan Dreja chanced to meet Top Topham, and introduced Topham to his brother. Topham and Dreja were influenced by folk/blues guitarist Gerry Lochran; he influenced them to switch from acoustic to electric guitars according to Greg Russo in his book The Yardbirds: The Ultimate Rave-Up. They made their debut with electric guitars at concert with Duster Bennett and a young Jimmy Page.
Dreja and Topham became core members of the Metropolitan (or Metropolis) Blues Quartet. During the space of a year Keith Relf, Jim McCarty, and Paul Samwell-Smith joined the group which became the Yardbirds. The 15-year-old Topham left the group when the band went professional but Dreja continued on to play rhythm guitar with musicians such as Eric Clapton and later Jeff Beck.
Dreja changed from rhythm to bass guitar following the departure of the original bassist, Paul Samwell-Smith. Dreja co-authored many Yardbirds group compositions, especially those on the album Roger the Engineer.
After the group broke up, Page offered Dreja the position of bassist in a new band he was forming (later to become Led Zeppelin). Dreja declined in order to pursue a profession in photography. He photographed Led Zeppelin for the back cover of their debut album.
Dreja played in the Yardbirds spin-off band Box of Frogs in the 1980s, and had been part of the Yardbirds’ reformation from 1992 to 2013
In 2002, The Yardbirds re-emerged and a new album, Birdland, was released.
Dreja suffered a series of strokes in 2012/2013 and had not performed with The Yardbirds since mid-2012. In July 2013 it was announced that he had officially left the band for medical reasons and was replaced by original lead guitarist Top Topham.