Alternative, more commonly accepted parlance for "currentness"

The standard word is "currency". It doesn't just refer to money.

(In fact, the usage to refer to money comes from certain coinage being "current" in a certain locale, i.e. commonly accepted. Accordingly, Scottish notes, although not legal tender in England, are current.)

As Marcin notes, currency is usable here; one could say:

Ensure the currency of X.

However, the following phrasings are more likely:

Ensure that X is up to date. [or up-to-date]
Ensure that X is the latest version.
Ensure that X is current.

(where the latter is one of your examples.) Current as in current events usually has the sense "Existing or occurring at the moment", meaning now. Contemporaneous, on the other hand, means "Existing or created in the same period of time". For example:

Contemporaneous notes are preferred for documentation of incidents.

Some alternatives could be:

Ensure that X is up-to-date.

or

Ensure that X is updated.