We propose and analyze two agent-based models to investigate the dynamics of papal conclaves, focusing on how social influence, strategic voting, and ideological alignment affect the time required to elect a pope. In the first model, cardinals interact through two mechanisms: with probability $p$, they imitate the choice of a randomly selected peer, and with probability $q$, they shift support to the most voted candidate from the previous round. Additionally, strategic behavior is introduced via ``useful voting'', where agents abandon their preferred candidate if he receives less than a threshold fraction of the votes, switching instead to the most viable alternative. A candidate must secure a qualified majority of two-thirds to be elected. We then extend the framework by incorporating ideological blocs, assigning each cardinal and candidate to one of two groups (e.g., progressives and conservatives). Cardinals initially vote for candidates from their own group but may cross ideological lines for strategic reasons. We initialize the electorate with $20\%$ conservative cardinals, reflecting the current composition shaped by papal appointments. Numerical simulations show that ideological polarization tends to delay the election by increasing the number of voting rounds required. However, higher values of strategic responsiveness $q$ can restore efficiency even under polarization. We further validate the model by calibrating parameters to historical data from conclaves held between 1939 and 2025. The model reproduces observed convergence times with good agreement, supporting its explanatory power across institutional contexts. The rapid outcome of the 2025 conclave, despite ideological divisions, suggests the importance of informal consensus-building, possibly prior to voting, as a key mechanism for accelerating convergence.