London is one of the oldest cities in the world; it's incredibly difficult to dig down and not unearth old gravesites/cemeteries or random artifacts. The Crossrail Project back in 2009/2010, for example dug up over 10,000 artifacts while laying down new rail expansions for the Tube, enough that the Museum of London Docklands opened up an entire exhibit dedicated to them. The same project unearthed the Bedlam Burial Grounds, old Roman coins, broken Victorian glassware, medieval ice skates, a Tudor-era bowling ball, and gravesites for Black Plague victims. It is not an understatement to say that archeologists, engineers, and construction workers are literally always finding things whenever they dig.

This National Geographic article examines some of the many millions of artifacts found under London over the years. Specifically regarding the 20th century, what you really are looking for is finds unearthed by the creation of the Victoria and Jubilee lines and various extensions made to the already existing lines. Specifically relating to the creation of the Victoria Line in the 1960s, miners and construction workers unearthed everything from fossilised marine mollusks to human bones, and the construction of the Jubilee Line in the 70s found everything from a Roman building during the building of London Bridge Station to a Neolithic arrowhead at Westminster to a medieval monk's cemetery at Stratford. A lot of the information you're looking for is provided by the Museum of London Archeology, who put out books like The Big Dig and London's Archeological Secrets: A World City Revealed specifically chronicling the finds from said construction efforts.

In regards to people reporting archeological finds incorrectly, the BBC ran a piece a while back debunking some of the claims that Tube lines were designed to avoid plague pits and talks about some of the various archeological finds that actually did happen during the construction of the Underground, including old church graveyards. Because it's London, it's essentially impossible to dig down and not find human remains at some point. However, the urban legend of plague pits and mass gravesites being uncovered "all the time" during construction is deeply exaggerated. Anyway, that article is fairly extensive in chronicling the finding (or not finding) of human remains during construction of the Underground.