Composite-overwrapped pressure vessels are essential for high-pressure hydrogen storage in the automotive and energy sectors, yet safe and cost-effective storage remains a major challenge. Analytical models determining stress fields are extremely useful for design optimisation and parameter studies, especially in initial development stages. The prevailing technical netting theory, which is based solely on membrane stresses, is inadequate since it neglects the bending stresses which can exceed their membrane counterparts in Type IV pressure vessels. The present contribution introduces a novel closed-form analytical model for predicting the stress concentrations due to discontinuity loads at the cylinder-dome junction of thick laminated composite pressure vessels. The approach, formulated using a layerwise stress function-based approach, is applicable to arbitrary asymmetric laminate layups and captures interlaminar stress concentrations in thick composite shells of both geometries. Exploiting the axisymmetric loading nature of internal pressure, the vessel is idealised as a two-dimensional structure subjected to a uniform edge bending moment load and transverse load at the cylinder-dome junction. The far-field solutions for the respective geometries are established by defining unknown stress shape functions through a layerwise plane strain formulation, while the localised stress concentrations at the geometric discontinuity are represented by a characteristic eigenvalue solution where the decay parameters are the roots of an analytically-derived characteristic equation. Complete 3D stress and displacement fields are obtained by enforcing layerwise compatibility at the junction, enabling the treatment of mixed boundary conditions. Existing closed-form analytical models are either valid only for the cylindrical region, or only for symmetric laminate layups ignoring out-of-plane stresses. In contrast, the current methodology is formed within a unified framework and is valid for generic laminate layup sequences. Verifications against equivalent finite element simulations show excellent agreement across the entire stress field. For an internal pressure load of 70MPa, the results observe transverse shear stress fields reaching critical values upto 60MPa at the junction, thus necessitating the importance of capturing the discontinuity stresses.