A perfectionist is defined as follows:
a person who refuses to accept any standard short of perfection.
But I am looking for a noun which describes a person who refuses to move forward in an activity or thought unless the present one is sorted out. Kind of like having a mental block to move ahead, but by choice.
It's not listed in the major dictionaries, but Wiktionary mentions completionist:
One who insists on completion.
I wouldn't count on your audience knowing exactly what it means, so you might have to explain it first, but you probably wouldn't get any closer with a single word.
I would use single-minded (Merriam)
: having one driving purpose or resolve : DETERMINED, DEDICATED
or unitasker (wiktionary)
A person who does a single thing at a time.
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.
Single-threaded (a term borrowed from computer science)
In computer programming, single-threading is the processing of one command at a time. The opposite of single-threading is multithreading. Thread (computing) - Wikipedia https://en.wikipedia.org/wiki/Thread_(computing)
"He had a single-threaded mind"
You could use Linear, according to Merriam Webster Dictionary:
of, relating to, or based or depending on sequential development
Sticktoitiveness (Grammarist):
Sticktoitiveness — meaning dogged perseverance — regularly appears in three forms: the unhyphenated form, stick-to-it-iveness, and stick-to-itiveness. The three-hyphened form was most common when the word came about in the U.S. in the late 19th century (as attested by the OED‘s examples and in historical news searches). But today, the two-hyphened stick-to-itiveness is most common, and the unhyphenated form is still comparatively rare. The last form is steadily gaining ground, though, and is likely to prevail in the near future, if the word stays in the language.
If the behavior is particularly extreme or obsessive, you might call them monomaniacal. According to Merriam-Webster:
monomania: excessive concentration on a single object or idea
You could say that someone is "mired" in a task - a mire is like a swamp or bog, so it's a metaphor for being stuck in something. This doesn't necessarily mean that they wouldn't like to leave it and do something else, but it may often be the case that they are refusing to move on.
First word that came to mind is persistent (from Wiktionary, 2019):
Obstinately refusing to give up or let go.
This means that when someone is engaging in a task there is a reluctance to let go of performing the task, even when not appropriate anymore.
From Cohen & Levesque (1990):
It does not seem unreasonable to require that a robot not procrastinate forever. Moreover, we surely would want a robot to be persistent in pursuing its goals, but not fanatically so. Furthermore, we would want a robot to drop goals given to it by other agents when it determines the goals need not be achieved.
Some personality tests use the term completer-finisher alternatively just completer or finisher. An example personality test is Myers Briggs, although some other personality tests use these terms too. A completer-finisher goes the extra mile to make sure that everything is just right and can frustrate their team mates at work by wanting to get everything done exactly right before moving on to the next task.
fixated
Google dictionary (for fixate):
cause (someone) to acquire an obsessive attachment to someone or something.
When I read the title of your question 'Single word for “refusing to move to next activity unless present one is completed.”' I imdieately thought of 'go/no go'
From Wikipedia
In general go/no go testing refers to a pass/fail test (or check) principle using two boundary conditions or a binary classification. The test is passed only when the Go condition is met and also the No go condition fails.
But it does not meet several of your criteria, it is not a noun, it does not describe a person, and it is not a single word.
In my experience (US Army) it is often used like a single word to describe where the only options are move forward or not.
While it might not be exactly what you are looking for, it may provide alternate usage in the same context.
Depending on the context, you may be able to use prerequisite as a reference to the thought or process itself rather then the person. A prerequisite of future tasks may not be quite what you are looking for, but it is a different perspective to consider.
You could also describe someone as being fixated on a task. This seems to capture the sense you're looking for almost perfectly. But if you're describing a long-term personality trait rather than describing an immediate behavior, it might not work as well.
