The map $\widetilde{T} \rightarrow T$ is an isogeny, and an isogeny between tori over any field $K$ induces a finite-index inclusion between the associated geometric character groups as ${\rm{Gal}}(K_s/K)$-lattices, hence an isomorphism on associated rational vector space, so same dimension for spaces of Galois-invariants; i.e., same rank for $K$-rational character groups. However, this is a central isogeny, and that is really the key to the good behavior.
For isogenies $f:G \rightarrow G'$ between connected semisimple $K$-groups such that $\ker f$ is not central in $G$ (which can only occur in positive characteristic $p$, such as Frobenius isogenies $F_{G/K}:G \rightarrow G^{(p)}$) it can happen that $G'$ has larger $K$-rank than $G$. This might sound paradoxical if you are not familiar with it, since you might reason that if $T' \subset G'$ is a maximal $K$-torus (say containing a maximal split $K$-torus of $G'$) then the identity component of the underlying reduced scheme of $f^{-1}(T')$ seems to be a smooth $K$-subgroup scheme of $G$, hence a torus mapping onto $T'$ via an isogeny, so it has the same $K$-rank as $T'$ by the first paragraph above. That reasoning is valid provided that $f^{-1}(T')_{\rm{red}}$ really is a smooth $K$-subgroup scheme of $G$. For perfect $K$ such conditions hold (since the underlying reduced scheme of a finite type $K$-group scheme is a smooth $K$-subgroup scheme for such $K$). But it can fail when $K$ is imperfect.
For example, if $K$ is a local function field of characteristic $p$ and $A$ is a central division algebra of dimension $p^2$ over $K$ then $G := {\rm{SL}}_1(A)$ is a $K$-anisotropic form of ${\rm{SL}}_p$ but $G^{(p)}$ is $K$-split since $A^{(p)}$ is $K$-split (by local class field theory). So $F_{G/K}:G \rightarrow G^{(p)}$ is an isogeny from a $K$-anisotropic absolutely simple semisimple $K$-group onto a $K$-split one. And of course this is a non-central isogeny. I suspect that this is the kind of phenomenon you were trying to find when formulating the OP.