Charge transfer in conductors of different metals by touching
Charge transfer accompanying bringing into contact (and then possible separation) of two metallic bodies is a multifaceted physical phenomenon. John Lekner studied a process of equalization of galvani potentials and published a notable research in Proc. of the Royal Society A, 'Electrostatics of two charged conducting spheres'.
If the bodies are made of different metals, bringing these into contact equalizes their electrochemical potentials (fermi levels), see Steve Byrnes' answer to a question in physics SE 'Fermi level alignment and electrochemical potential between two metals'.
Anyway, to calculate time dependent transient currents of the charge transfer process [in order to determine &quot;time constants (charging time) in both cases&quot;], you need to solve a PDE system with parameters and boundary/initial conditions specific for geometries and physical properties of the bodies you study.