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Back before electronics took over calculations, a calculator was a person's job title (and you needed skill to do that). What's that skill called?

A term for someone who can juggle many numbers, routes, and other information in their head at the same time, while re-arranging / re-assigning them to different workers.

What is your source for the claim that such a job title was 'calculator'? My first job (in the mid-1960s) was at a very distinguished scientific research institution (5 Nobel prize winners on the staff at the time - and more to follow.)

My job was in the computer department. I was a temporary assistant computer, my permanent colleagues (female) were computers, and our boss was the chief computer.

There was a rather eccentric mathematician, in another department, who had an 'electronic computer' and part of my job was to take boxes of punch cards to a big electronic computer in another city.

Your description of the duties of the person you want to call a calculator is in line with those of the head of the computer department in my lab at that time. (She had graduated in mathematics, with high honours in a distinguished university).

There is no such job in the modern world, and therefore no job title that describes it in modern terms. The answer to your question then is that no such word exists.

In our modern admiration for all the things that electronic computing can do for us we should not forget the skills of such people as clerks in the Post Office who could add up a column of figures at a glance, or vehicle schedulers who could sort out in their heads a very good, maybe not universally the best, route to deliver ice cream to shops in awkward-shaped counties like Cornwall.

Wikipedia offers the term mental calculator:

Mental calculators are people with a prodigious ability in some area
of mental calculation, such as adding, subtracting, multiplying or
dividing large numbers.

Mental calculators were in great demand in research centers such as
CERN before the advent of modern electronic calculators and computers.

Human computer is another option:

The term "computer", in use from the early 17th century (the first
known written reference dates from 1613), meant "one who computes": a
person performing mathematical calculations, before electronic
computers became commercially available. "The human computer is
supposed to be following fixed rules; he has no authority to deviate
from them in any detail." Teams of people were frequently used to
undertake long and often tedious calculations; the work was divided so
that this could be done in parallel. Frequently, the same calculations
were performed independently by separate teams to check the
correctness of the results.

Since the end of the 20th century, the term "human computer" has also
been applied to individuals with prodigious powers of mental
arithmetic, also known as mental calculators.

In the old days before electronic calculators we used a comptometer.

It was used by a comptometer operator. This was a highly skilled task, a (then) modern version of the abacus. The operator's fingers would form a cradle of fixed shape depending on one operand and dance among the rows and columns to obtain a quick result from the other operand.

I also know the word wrangler which I understand to be someone in the days before electronic calculators, who was proficient in calculating values such as log tables. I can't find a better reference than this.

As answered above computer was an old term. Notable pioneers in this field included Mary Edwards (human computer)
Mary Edwards (c. 1750 – September 1815) was a human computer for the British Nautical Almanac Her daughter, Eliza Edwards (1779-1846), also worked as a computer [until...] rules made the employment of women very difficult.

Side note
The supposed first digital calculator was "The Arithmometer" or Arithmomètre not introduced till Patented in France by Thomas de Colmar in 1820 long after computers.

So for you quiz masters "Which came first the computer or the digital calculator?"

Personal experience.
My Great Aunts reputedly lived over the age of 100 and worked with the navy, in The Great War (WWI), but would never disclose what they did, however the family mumbled they were computers.

arithmetician vocabulary.com

n

someone who specializes in arithmetic

As in:

He was a natural mathematician, and was the most profound and original
arithmetician in the Southwest. Rhodes, W. H. (William Henry)

and

You are a clever arithmetician, mamma; you do your sums and get your
totals nicely. Dostoyevsky, Fyodor