~~Here's another example that I think should work, but in the p-adic world.~~

~~Let $F$ be a p-adic field, and consider the group $G = SL(2,F)$. Inside $G$, we various elliptic tori as follows : Let $E$ be a quadratic extension of $F$. Then $T := E^1$, the set of norm $1$ elements in $E$, embeds in $SL(2,F)$ as $a + b \delta$ maps to the $2 \times 2$ matrix $(a,b, b \Delta, a)$ where $E = F(\sqrt{\Delta})$, $\delta = \sqrt{\Delta}$.~~

~~i.e. a in upper left, b in upper right, $b \Delta$ in lower left, $a$ in lower right (sorry, I don't seem to be using the array command correctly here).~~

~~Now, $G$ has a canonical $2$-fold cover $\widetilde{G} = \widetilde{SL(2,F)}$, the metaplectic cover. It sits in an exact sequence $$1 \rightarrow \mathbb{Z} / 2 \mathbb{Z} \rightarrow \widetilde{G} \rightarrow G \rightarrow 1$$
This is a topological central extension. Moreover, the 2-cocycle of this extension can be written down explicitly, but we won't need the full cocycle. For more information, see page 7 of ~~

~~http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=7B8F219E349E46B736984D32600EB0CB?doi=10.1.1.78.7539&rep=rep1&type=pdf~~

~~This extension restricts to an extension $$1 \rightarrow \mathbb{Z} / 2 \mathbb{Z} \rightarrow \widetilde{T} \rightarrow T \rightarrow 1$$
where $\widetilde{T} := \pi^{-1}(T)$, where $\pi$ is the projection $\pi : \widetilde{G} \rightarrow G$. It's a fact that the 2-cocycle of this extension is given by $c(x,y) = (x,y)_F$, where $( \cdot, \cdot)_F$ is the Hilbert symbol of $F$ (see loc. cit.)~~

~~In order to determine whether this extension splits as abstract groups, we just need to show whether any primage of $-1 \in T$ has order $2$ (for the proof of this, see page 8 loc. cit.). The definition of multiplication in an extension says that $(\pm 1, -1)^2 = (1, (-1,-1)_F)$. So at least if $F$ has residual characteristic $\neq 2$, we have that $(-1,-1)_F = 1$. Therefore, the above sequence splits as abstract groups.~~

~~However, by Remark 4.1 of loc. cit., this sequence also splits topologically.~~

After reading Kevin's comment, here is a response that hopefully will answer the question.

A general class of examples that answer the question can be gleamed from Moore's paper "Group extensions of p-adic and adelic linear groups". In this paper, Moore defined cohomology groups that take into account topology. That is, (I quote from the second paragraph of his paper) "If $G$ and $A$ are locally compact separable topological groups, and if $G$ acts on $A$ as a topological transformation group of automorphisms, one may modify the definitions and arrive at cohomology groups $H^n(G,A)$ which take into account the topology".

Let $G$ be a locally compact separable group, and $A$ a locally compact separable topological $G$-module. As in Moore's notation, we let $G^a$ and $A^a$ denote the underlying abstract groups of $G$ and $A$ respectively, considered without their topologies. Denote $H^n(G^a,A)$ the ordinary group cohomology (no topologies considered. Again this is Moore's notation). Then we have the natural homomorphism $$ H^n(G,A) \rightarrow H^n(G^a,A)$$

$\mathbf{Theorem \ 2.3}$ (of Moore's paper) : If $G$ is perfect, then the natural map $H^2(G,A) \rightarrow H^2(G^a,A)$ is injective.

Thus, with the given assumptions on $G$ and $A$, take any topological extension $$1 \rightarrow A \rightarrow E \rightarrow G \rightarrow 1$$ that splits algebraically. Then it splits topologically.

notimply that $G=N\times H$ as groups, and the semi-direct product given in my answer shows you this. Could you please add an edit to your question, making it clear exactly what you mean by "split exact sequence of topological groups" $\endgroup$1more comment