How to choose between "work day" vs "working day"

In general, a work day is a day on which you work, while the working day is that part of the day when you're at work: "my work days are Monday to Friday: at the end of the working day I go straight home to dinner".

Working hours can be used to be more specific: "working hours are 9-5". A total number of working hours per day is often used in flexible working systems with core hours, i.e. hours when everyone is expected to be in: "working hours: 37.5 per week, core hours 10-4" (typical in a job specification). Work hours isn't as common but means the same.

Business hours, as the name suggests, apply to the business rather than its staff, like opening hours for a shop; they are likely to be longer than any individual's working hours.

Some of this may be biased towards British usage, but not intentionally.

The answer is that they are virtually interchangeable.

However in composite nouns involving adjectival "...ing words" there is a tendency for Americans to drop in "...ing" part. Hence in the USA you hear them talk about fry pans (UK frying pans), swim shorts (UK swimming shorts), file cabinets (UK filing cabinets), a lead role (UK leading role) and hence work day (UK working day).

I think it is largely a difference between American and British treatment.

A quick visit to Merriam-Webster's Eleventh Collegiate Dictionary (2003) yields the following entries:

workday n (13c) 1 : a day on which work is performed as distinguished from a day off 2 : the period of time in a day during which work is performed — workday adj

...

working day n (15c) : WORKDAY

In other words, according to this general-reference resource, the noun workday and the noun phrase working day are interchangeable, meaning that both definitions that MW gives for workday also apply to working day.

MW asserts that workday is roughly two centuries older than working day, but both terms have been in use for more than 500 years. Workday may be more firmly established than working day as an adjective, but the absence of "— working day adj" from the dictionary's entry for working day should not, I think, be taken as a denial that working day can appear as a modifier in expressions such as "working day routine"; indeed, a Google Books search finds multiple instances of that particular phrase, sometimes with the modifier hyphenated (as "working-day routine") and sometimes with it unhyphenated (as "working day routine").

The Ngram chart for "working day" (blue line) versus "work day" (red line) versus "workday" (green line) for the period 1550–2019 shows a fair amount of variability in the various terms' popularity over the years:

Most notably, working day enjoyed a much more dramatic increase in frequency of use from about 1820 until about 1920 than did work day and workday, but in recent decades its advantage over workday has vanished.