Do the efficiency savings of mergers indicate a type of waste or inefficiency in a market? In this BBC News article about the merger of Tata Steel and Thyssenkrupp, the following figure is reported:


  The merged group anticipates it will make costs savings of between £350m to £440m a year.


Presumably some or all of these savings would be achieved by eliminating redundant duplication of departments across the two organizations.

Are the cost savings mentioned in the article indicative of a fundamental inefficiency caused by a competitive market, that wouldn't be present in, say, a nationalized industry (or total monopoly)?
Are the cost savings mentioned in the article indicative of a
  fundamental inefficiency caused by a competitive market, that wouldn't
  be present in, say, a nationalized industry (or total monopoly)?


That "causation" is mischaracterized. It is not that a competitive market causes inefficiencies, but that it encourages their removal. Inefficiencies also exist in a state-controlled economy, but there the government pursues other policies because it does not need to be concerned about the risk of being left out of the [state-controlled] market.

The title of the article you shared serves to illustrate the point. The fact that the unions "welcomed" the merger reflects how they typically object to mergers (elsewhere, the article quotes one union officer in terms of having "made great sacrifices"). The historical friction from unions is antithetical to the optimization that eventually becomes indispensable in a market-based economy. Here, it appears from the article that the unions understood that Trump's tariffs on steels and the imminent consequences from Brexit are compelling reasons to reach an agreement and accept the merger.

A natural monopoly is the only monopoly that could thrive in a competitive market because, by definition of natural monopoly, the existence of competitors would not result in improvements in the market or economy.