Pressure-driven pipe flow of two upper-convected Maxwell liquids in a vertical core-annular arrangement is studied. The annulus and core liquids have different relaxation times and viscosities. Weakly nonlinear evolution equations which describe the interface shape are derived. Lubrication theory is used in the annulus but not in the core. Motions periodic in the streamwise direction are addressed, with the aim of describing short-time behavior driven by capillary forces. Numerical and analytical results for the spatio-temporal dynamics are given for two subcases. In the first case, the liquids have the same viscosity. White noise non-axisymmetric initial data are found to evolve into axisymmetric motion. When axisymmetry is assumed, the evolution equation is a Kuramoto-Sivashinsky equation; the bifurcation parameter depends on the fluid elasticities, interfacial tension, Reynolds number and Weissenberg number. The second case concerns axisymmetric motions with wavelengths in the axial direction that are long compared with the annulus thickness. This asymptotic analysis places a restriction on the size of the Weissenberg number. A jump in the viscosities introduces dispersion, which may be enhanced by fluid elasticity; this can lead to a transition from an unstable regime or a chaotic regime to one in which organized traveling wave pulses move in the axial direction.