We investigate uniform ergodic type theorems for almost additive and subadditive functions on a subshift over a finite alphabet. We show that every uniquely ergodic subshift admits a uniform ergodic theorem for Banach-space-valued almost additive functions. We then give a necessary and sufficient condition on a minimal subshift to allow for a uniform subadditive ergodic theorem. This provides, in particular, a sufficient condition for unique ergodicity.