We introduce and study single-conclusioned nested sequent calculi for a broad class of intuitionistic multi-modal logics known as intuitionistic grammar logics (IGLs). These logics serve as the intuitionistic counterparts of classical grammar logics, and subsume standard intuitionistic modal and tense logics, including IK and IKt extended with combinations of the T, B, 4, 5, and D axioms. We analyze fundamental invertibility and admissibility properties of our calculi and introduce a novel structural rule, called the shift rule, which unifies standard structural rules arising from modal frame conditions into a single rule. This rule enables a purely syntactic proof of cut-admissibility that is uniform over all IGLs, and yields completeness of our nested calculi as a corollary. Finally, we define an interpolation algorithm that operates over single-conclusioned nested sequent proofs. This gives constructive proofs of both the Lyndon interpolation property (LIP) and Beth definability property (BDP) for all IGLs and for all intuitionistic modal and tense logics they subsume. To the best of the author's knowledge, this style of interpolation algorithm (that acts on single-conclusioned nested sequent proofs) and the resulting LIP and BDP results are new.