The profession plumber long predates English, appearing in Classical Latin as plumbārius. The Latin profession name is derived from plumbum, meaning “lead”. (You can still see this in lead’s elemental abbreviation Pb.) We’re not certain where plumbum itself comes from. Historically, it was attributed to Ancient Greek μόλυβδος (mólubdos), meaning “lead”. There’s a contemporary theory that it’s instead a sibling of Proto-Celtic ɸloudom (Old Irish lúaide, Modern Irish luaidhe), meaning “lead”. If that’s the case, the specific mechanics of the sound changes involved suggest there’s an unspecified common language both words were borrowed from.
For most of history, a plumber was a metalworker who worked with lead. Lead is both heavy and malleable at low temperatures. In the Roman Empire, refining galena ore was the primary source of silver. This produced lead as a byproduct, so it was also plentiful. All of those properties led to lead becoming the primary material for making pipes, roofing, and cookware. After the fall of Rome, lead was primarily known for its density. A plumb line is a lead weight on the end of a string used to measure if something was exactly vertical (aligned with gravity). Plumb is another word English acquired through Norman French where both the pronunciation and spelling dropped the ‘b’, but Latin scholars reinserted it into the spelling in the 1400s to be more authentic.
Lead pipes continued to be the most commonly used material through the rise of indoor plumbing in the late 1800s. Plumbing referring to interior piping infrastructure is from around 1875, not being common enough to warrant a word before then. Beginning around 1900, plumbers exclusively focused on interior piping. Around 1936, copper became the preferred material for pipes. Lead pipe was finally banned once the health risks became undeniable (1970 in the UK; 1986 in the US; 1995 in France).