Rigorous uncertainty quantification is essential for the safe deployment of autonomous systems in unconstrained environments. Conformal Prediction (CP) provides a distribution-free framework for this task, yet its standard formulations rely on exchangeability assumptions that are violated by the distribution shifts inherent in real-world robotics. Existing online CP methods maintain target coverage by adaptively scaling the conformal threshold, but typically employ a static nonconformity score function. We show that this fixed geometry leads to highly conservative, volume-inefficient prediction regions when environments undergo structural shifts. To address this, we propose $\textbf{AdaptNC}$, a framework for the joint online adaptation of both the nonconformity score parameters and the conformal threshold. AdaptNC leverages an adaptive reweighting scheme to optimize score functions, and introduces a replay buffer mechanism to mitigate the coverage instability that occurs during score transitions. We evaluate AdaptNC on diverse robotic benchmarks involving multi-agent policy changes, environmental changes and sensor degradation. Our results demonstrate that AdaptNC significantly reduces prediction region volume compared to state-of-the-art threshold-only baselines while maintaining target coverage levels.