Serialist could have something of the meaning you want. The problem is that it has some specific meanings in other contexts, and doesn't seem to have been used in relation to the completion of tasks. The closest meaning in the OED is meaning #4, but it relates more to learning rather than the completion of tasks.
A person who writes stories, novels, etc., for publication in serial form.
Philosophy. A believer in or advocate of a theory based on the analysis of the self as a series or succession of states or events. rare.
Music. A composer, advocate, or admirer of serial music or composition.
Psychology. A person who tends to acquire knowledge about something by consideration of a series or sequence of items, facts, etc., in turn.
Examples given for meaning 4 are:
"Serialists learn, remember and recapitulate a body of information in small, well-defined and sequentially-ordered segments."
"Operation learning is the style of those who are routine serialists; comprehension learning that of routine holists."
More broadly, you could say that such a person is "stubbornly serial" or "doggedly serial" in completing their tasks, though you'd probably have to add a subordinate clause explaining what you mean.