  We report an intriguing transition from the quantum spin Hall phase to the
spin Hall effect upon segregation of thallium adatoms adsorbed onto a graphene
surface. Landauer-B\"uttiker and Kubo-Greenwood simulations are used to access
both edge and bulk transport physics in disordered thallium-functionalized
graphene systems of realistic sizes. Our findings not only quantify the
detrimental effects of adatom clustering in the formation of the topological
state, but also provide evidence for the emergence of spin accumulation at
opposite sample edges driven by spin-dependent scattering induced by thallium
islands, which eventually results in a minimum bulk conductivity $\sim
4e^{2}/h$, insensitive to localization effects.
