。
The deadline is next Monday or Tuesday.
Literal
Next-week [possessive-の] Monday [or-か] Tuesday [until-まで] [explanatory-なんだ]-but.
The particle か between nouns = 'or' (Monday or Tuesday). なんだけど ends the sentence with an explanatory-softening preface — the speaker is laying out context but leaves the conclusion hanging ('...and that's the situation, but...'). This ending invites the listener to respond.