Learn English – When/what is the expiration date

Neither of OP's alternatives are acceptable compressed forms of what would normally be referred to as date of starting work in the full form of this compound noun. The most common short form is...

work start date

Note that the base noun here is date - modified by another noun start, which is itself modified by the noun work. Most compound nouns involve only two nouns anyway, and if there are more than two they usually all modify the same base noun, but the only grammar involved here is that when a noun is being used "adjectivally" it follows the normal rule for adjectives - it comes immediately before the noun it modifies (even if that modified noun is then used adjectivally to modify another noun that follows).

In American English, we don't normally write dates that way, but instead write and say with the month first — even if it's written day first, we'd still usually pronounce it month first. So "27 July" becomes "July twenty-seventh".

I believe other dialects usually, but not always, include the article before the day when putting the day before the month, so "[the] twenty-seventh [of] July". It will be understandable no matter which way you say it, but it's safer to use the slightly longer form to avoid sounding awkward in places that aren't that terse.