Time-resolved simultaneous synchrotron small-angle and wide-angle X-ray scattering (SAXS and WAXS) and DSC experiments have been performed on poly(ethylene)-poly(ethylethylene) and poly(ethylene)-poly(ethylene-propylene) diblock copolymers quenched from melts with lamellar and hexagonal-packed cylinder structures. We find that the original microphase-separated morphologies are destroyed due to poly(ethylene) (PE) chain folding upon crystallization. Below the melting temperature, a sample with a volume fraction f(PE) = 0.49 forms a lamellar structure distinct from that in the melt. On quenching, a hexagonal f(PE) = 0.25 sample forms a lamellar structure similar to that of the f(PE) = 0.49 sample, while the hexagonal-packed cylinder structure of an f(PE) = 0.75 sample is also destroyed by crystallization but does not form well-ordered lamellae. The SAXS profiles from crystallized materials are shown to correspond to the sum of scattering from block copolymer lamellae, with up to four orders of reflection, plus a broad peak arising from semicrystalline PE. Analysis of scattering density correlation functions calculated using the SAXS data shows that the PE lamellar thickness is (45 +/- 5) Angstrom for all samples, similar to that observed for PE homopolymer. The WAXS data reveal that PE crystallizes in its usual orthorhombic form in all samples. The relative degree of crystallinity as a function of time after a quench, determined from the SAXS invariant, is fitted by Avrami equations for spherulitic crystallite growth. The Avrami exponent is found to be n = (3.0 +/- 0.1) for all samples, consistent with a nucleation and growth process.