The definition is a bit misleading as syndetic subgroups are not assumed to be normal, but you require them to be normal in defining "syndetically separated". So let me change the definition, removing the word "normal" in the definition of "syndetically separated", turning your original definition into "normally syndetically separated".

1) If $G$ is a linear Lie group with a cocompact lattice, then it is syndetically separated (in my sense). Indeed, if $\Gamma$ is a cocompact lattice, then $\Gamma$ is residually finite, in the sense that the intersection of all finite index subgroups is trivial. If $K$ is a compact subset of $G-\{1\}$, then $K\cap\Gamma$ is finite and it follows that some finite index subgroup $\Gamma_1$ of $\Gamma$ satisfies $\Gamma_1\cap K=\emptyset$.

2) The simplest noncompact example of such a group $G$ is just the additive group $\mathbf{R}$ of real numbers and more generally $\mathbf{R}^n$. These groups are therefore normally syndetically separated (this answers your question). Also, residually finite discrete abelian groups are normally syndetically separated, e.g. $\mathbf{Z}[1/p]$.

3) Other examples satisfying (1) are linear connected semisimple Lie groups, as well as $p$-adic analogues. On the other hand, if $G$ is a nontrivial connected Lie group whose Lie algebra does not have any quotient abelian or compact simple (e.g. $G$ is simple and noncompact, e.g. $\text{SL}_2(\mathbf{R}))$, then the only normal cocompact (=syndetic) subgroup of $G$ is $G$ itself and therefore $G$ is not normally syndetically separated.

4) More generally, a connected Lie group $G$ is normally syndetically separated iff its Lie algebra is product of an abelian one and a compact semisimple one. Equivalently, $G$, is the direct product (up to finite covering) of a compact semisimple Lie group and an abelian Lie group. Indeed, if $\mathfrak{n}$ is the intersection of the kernels of all homomorphism from $\mathfrak{g}$ to abelian or compact simple Lie algebras, then it can be shown that the Lie subgroup $N$ it generates is closed and equal to the intersection of all kernels of homomorphisms from $G$ to compact Lie groups (I skip the proof). Example: the Heinsenberg group is not normally syndetically separated.

5) In view of (1), it is natural to wonder about the existence of a connected Lie group which is *not* syndetically separated (in my sense). An example is the affine ``ax+b" group $\mathbf{R}\rtimes\mathbf{R}$. Indeed it is not hard to show that any proper closed cocompact subgroup (not assumed normal!) has the form, in the above decomposition $\mathbf{R}\rtimes \lambda\mathbf{Z}$.

6) $\mathbf{Q}_p^n$ for $n\ge 1$ is not syndetically separated (it has no proper cocompact subgroup).