Is there a hypernym for or construct for "get current or next"

Effective may be an effective hypernym for your purpose.
Effective date is typically used in business contexts to signify a date when something comes into effect. In your context, effective would mean current or (if current is not applicable) next (business day).
For example, if you give your bank an instruction (electronically or otherwise) during non-business hours (like on a Saturday or sometimes even after business hours on a working day), usually they are not bound to execute it until the nearest available business hours, typically due to "operational" reasons. (There may be exceptions to this.)
Also, usually in such contexts, the accounting entry will have the date of the next working day.

Wikipedia:

An effective date or as of date is the date upon which something is considered to take effect, which may be a past, present or future date. This may be different from the date upon which the event actually occurs or is recorded.

Suggestion 1. just: get the working day, getWorkingDay.

On Friday, this is Friday. On Saturday, this is Monday.

In many English language situations, the way to specify "the current or next" is to specify nothing; use the or a bare subject.

So: "get the newspaper" -- get the currently available or next available newspaper, whether already at-hand or from a newsstand.

Suggestion 2: If "getWorkingDay" seems potentially ambiguous (perhaps the working day is "None"?), use a proximity word like 'nearest' or 'closest' which will always include the current day if appropriate:

getNearestWorkingDay()
getClosestWorkingDay()
getProximalWorkingDay()