Yes, rings as any algebraic theory make a locally presentable category, and this is well- copowred (Adámek and Rosicky, Locally Presentable and Accessible Categories Cambridge University Press, Cambridge, (1994)) 

About topological case, in the MAria Clementino article "Categorical and topological aspects
of semi-abelian theories" (http://www.math.yorku.ca/~tholen/HB07BournClementino.pdf)
ther is a call from a Wyler article (see  10.1 in Clementino article), from this follow that: the forgetful funtor from  topological rings category to rings category preserve colimits (and these categories have colimits), the it preserve epimorphism, then from the well copoweredness of rings category follow the well cowellpowerness of the topological rings category

I seems (and hope) that this work well...
 
(Please, excuse my porr English).