Let's color each edge red with probability $p$, independently of other edges. Two vertices of $G$ glue together if and only if they are connected with a red path. Hence, your $V(G')$ is exactly the number of connected components of the subgraph, obtained from $G$ by keeping the red edges and removing all other (non-colored) edges. That is, you are interested in the expected number of connected components in the random subgraph of $G$, obtained by deleting every edge of $G$, randomly and independently from other edges, with probability $q:=1-p$. This quantity has been studied: see, for instance, this paper by Alon, or just Googlegoogle for something like "connected components of a random subgraph".
To complement the answer above, here is a simple lower bound which, intuition suggests, may be reasonably sharp for a wide class of graphs.
For a given vertex $v$, the probability that $v$ gets disconnected from the rest of the graph is $q^{d(v)}$ (where $d(v)$ denotes the degree of $v$). Therefore, the expected number of connected components of $G'$ is at least $$ \sum_v q^{d(v)}. $$ Now, let $\bar d$ denote the average degree of $G$. Since $G$ has at least $n/2$ vertices of degree at most $2\bar d$, the expected number of connected components is $\Omega(nq^{2\bar d})$. Thus, we have, say,
$$ {\mathsf E}(V(G')) = \Omega(ne^{-3p {\bar d} }) $$
subject to some mild technical restriction (like $p<1/2$).