Could micrograms of polonium really change the viscosity of a whole teapot?
I think it is highly unlikely polonium makes water notably more viscous. The concentration is just too low, and radiation doesn't make water more viscous, controlrods are cooled using water and they produce a lot more radiation.

what can happen in the case of tea is that the radiation induces the formation of free radicals that in turn initiate polymerization or other reactions that form larger molecules which would make the solution more viscous, this still sounds highly improbable though.

there are multiple studies on radiation initiated polymerisation:
http://large.stanford.edu/courses/2012/ph241/weil1/