This article explores the impacts of thermal radiation and viscous dissipation on the magnetohydrodynamic (MHD) bioconvection flow of a novel type of water-based upper-convected Maxwell nanoliquid containing nanoparticles and motile gyrotactic microorganisms over an absorptive vertical moving plate. The bioconvection in the nanoliquid is generated through the combined influences of buoyant forces and a magnetic field on the interplay between motile microorganisms and nanoparticles. The governing nonlinear partial differential equations of the problem are transformed into a set of nonlinear ordinary differential equations using suitable similarity transformations and a shooting method approach. The comprehensive flow regime is analysed parametrically to illustrate the effects of key parameters, namely the bioconvection Lewis number (Lb), traditional Lewis number (Le), bioconvection Peclet number (Pe), buoyancy ratio (Nr), bioconvection Rayleigh number (Rb), Brownian motion parameter (Nb), thermophoresis parameter (Nt), Hartmann number (Ha), Grashof number (Gr), radiation parameter (R), Eckert number (Ec), microorganisms concentration variation parameter (Ω), and suction/injection parameter (fw) on the flow, temperature, volume fraction of nanoparticles, and density profiles of motile microorganisms.