However, if more information is known then this non-constructivity can be overcome. For example, with a martingale, if the limit is computable in the $L^1$ norm and the $L^1$-bound is computable, then the rate of convergence is computable (both convergence in measure and a.e. convergence). (I assume the proof is constructive, but I haven't worked that out. However, computable rates of convergence are one of the most important consequences of constructive proofs.)
Similarly, the Lebesgue differentiation theorem has computable rates of convergence ($L^1$ convergence and a.e. convergence) since the limit is known (it is the function itself) and the Hardy-Littlewood maximal lemma controls the maximal amount of deviation from the limit.