SARS-CoV-2 vaccines should induce broadly cross-reactive humoral and T cell responses to protect against emerging variants of concern (VOCs). Here, we inactivated the furin cleavage site (FCS) of spike expressed by a modified vaccinia Ankara (MVA) virus vaccine (MVA/SdFCS) and found that FCS inactivation markedly increased spike binding to human ACE2. After vaccination of mice, the MVA/SdFCS vaccine induced eightfold higher neutralizing antibodies compared with MVA/S, which expressed spike without FCS inactivation, and protected against the Beta variant. We next added nucleocapsid to the MVA/SdFCS vaccine (MVA/SdFCS-N) and tested its immunogenicity and efficacy via intramuscular (IM), buccal (BU), or sublingual (SL) routes in rhesus macaques. IM vaccination induced spike-specific IgG in serum and mucosae (nose, throat, lung, and rectum) that neutralized the homologous (WA-1/2020) and heterologous VOCs, including Delta, with minimal loss (<2-fold) of activity. IM vaccination also induced both spike- and nucleocapsid-specific CD4 and CD8 T cell responses in the blood. In contrast, the SL and BU vaccinations induced less spike-specific IgG in secretions and lower levels of polyfunctional IgG in serum compared with IM vaccination. After challenge with the SARS-CoV-2 Delta variant, the IM route induced robust protection, the BU route induced moderate protection, and the SL route induced no protection. Vaccine-induced neutralizing and non-neutralizing antibody effector functions positively correlated with protection, but only the effector functions correlated with early protection. Thus, IM vaccination with MVA/SdFCS-N vaccine elicited cross-reactive antibody and T cell responses, protecting against heterologous SARS-CoV-2 VOC more effectively than other routes of vaccination.