The transient free convection flow of nanofluids between two long vertical parallel plates has been investigated analytically in the presence of thermal radiation by considering prescribed wall temperature (PWT) and prescribed wall heat flux (PWHF) at one boundary while the other boundary is maintained at the initial fluid temperature. The exact analytical solutions for the nanofluid velocity, temperature, skin friction, and Nusselt number are derived in the form of rapidly converging series with the help of Laplace transform technique. Five different types of water-based nanofluids containing copper (Cu), silver (Ag), copper oxide (CuO), titanium oxide (TiO2), and aluminium oxide (Al2O3) are considered in the analysis. The effects of nanoparticle volume fraction, radiation parameter, and temporal variable on the velocity, temperature, skin friction, Nusselt number, volume flow rate, and vertical heat flux have been discussed in detail. Some new heat and fluid flow characteristics of nanofluids have been presented. The present results can be used as a benchmark to validate the numerical solutions of transient free convection flow of nanofluids in a vertical channel for limiting cases. Also, the results are useful in gaining a deeper insight into the relevant practical systems.