Cohomology and Ext for rank one finite groups of Lie type in cross characteristic

We compute the dimensions of ${\mathrm{Ext}}_G^n(V,W)$ for all irreducible $V$, $W$ lying in $r$-blocks of cyclic defect in the simple groups $\mathrm{Sz}(q)$, ${\mathrm{PSU}}_3(q)$ and ${\mathrm{}}_2(q)$ in cross characteristic, obtaining in particular the dimensions of all cohomology groups for such modules. Along the way, we also obtain an analogous result for any $r$-block of cyclic defect whose Brauer tree is either a star or line (open polygon).