Confined at the nanoscale level, polymers crystallize much slower than in bulk, and in some cases the formation of ordered structures results inhibited for extremely long experimental time scales. Here, we report on the thickness dependence of the cold crystallization of thin poly(L-lactide) (PLLA) films (<300 nm), capped between two aluminum (Al) layers. The crystallization kinetics was monitored by means of dielectric relaxation spectroscopy, following the reduction in dielectric strength during annealing in isothermal experiments. We exploited a recently developed analytical method assessing the impact of irreversible chain adsorption and permitting to disentangle finite size and interfacial effects. In line with previous literature, the conversion time increased upon reduction of the thickness and crystallization was inhibited in films thinner than 10 nm. Moreover, we analyzed the thickness dependence of the dielectric strength and obtained the gradient in segmental mobility inside our capped films. We conclude that irreversible adsorption of chains onto the Al electrodes ultimately leads to a reduction in molecular mobility compared to the bulk.