Hi
Danish speakers will have to correct me, but I am quite confident that what follows is correct.
I think the orthographic similarity may have misled you to think that the two expressions are interchangeable, but they have very different properties in terms of meaning, syntactic behaviour and pronunciation. Any confusion of the two made in speech will probably be ignored by Danish speakers. They will just interpret it according to what fits the context.
Meaning
Hver dag - is a quantified expression and the interpretation is context-sensitive. It doesn't necessarily pick out Monday to Sunday, it can easily change to any number of days given the context. "During my holidays in Spain, I went to the beach every day."
Hverdag - is a noun that denotes what is not holiday/weekend etc. The meaning is more connected to routines and fixed schedules. It is not a quantified expression so it does not pick out a specific number of days.
Syntax (adding to justous' comments)
Hver dag - can be used as a noun phrase or as an adverbial phrase. As a noun phrase, further modification is very restricted. It works with adjectives, but it is not possible to combine with definiteness, plural formation, possessives...
Hverdag - is a noun only and can in principle be combined with any modifier that is available to Danish nouns.
Pronunciation
What you call a nuance in pronunciation has contrastive effects for native speakers. My guess is that the stress properties are not the same. Moreover, I would guess that the distribution of the famous Danish stød plays a role in distinguishing the two as well but that is just a mere guess.
Hver dag - main stress fall on "dag"
Hverdag - main stress falls on "hver"
To sum up: in your example phrase, "Jeg spiser i kantinen hverdag/hver dag", Danish speakers would not process "hverdag/hver dag" as a noun (a noun is simply not licensed in that position) and they would go for the adverbial interpretation even if your pronunciation could suggest otherwise.