Doesn't MediaWiki offer for all Wikimedia wikis the necessary support for formating dates according to user preferences or local wiki defaults for signatures?
If so, the number 6 you want will be inappropriate in Arabic and in many date formats for South Asian languages using other digits.
MediaWiki offers {{formatnum:6}} for this purpose, it is easy to infer where it is in the translation if you intend to change it. However June is also the 6th month of the year (same day number) and you may wonder which field is the day part if you don't know if the date uses "d;m;y" or "m:d;y" formatting order (some languages don"t really have names for Gregorian dates, but append a speific suffix to numbers to indicate if this is a year, month or day, such as Chinese).
A very small script will generate the formated dates that are then easy to update in translations, even if you don't want to use templates for that (because templates are wiki-dependant and their name vary).
But things would be even easier if all wikis had a parserfunction for formatting dates given a matching language code (if the format is unknown for that language, just use the ISO 8601 format yyy-mm-dd which is not ambiguous in all existing languages, even if it is not the most usual format for that language.
I just hope that one day we'll have a parser function like {{formatdate:2014-06-06|ymd|language-code}} that will format the date using the 3 fields (not padded with zero if not needed, but not rounded to 2 digits with the appropriate order, suffixes and separators, and possibly their names; all in the appropriate script)
CLDR data has this info and there's even a demo in the Unicode CLDR website.