Let $\bar M_E = \varepsilon_* \mathbb{Z}$ where $\varepsilon: \tilde E_b\to E_b$ is the normalization map of the special fiber. If the irreducible components $X_1, \ldots, X_r$ of the special fiber are smooth (i.e. have no self-intersections) then $\bar M_E \cong \bigoplus_{i=1}^r \mathbb{Z}_{X_i}$.
Let $\bar M_{E/B}$ be the quotient of $\bar M_E$ by $\mathbb{Z}$ so that we have an extension $$ 0\to \mathbb{Z} \to \bar M_E \to \bar M_{E/B}\to 0. $$
Then there are canonical isomorphisms $$ \mathbb{Z} \cong \psi_f^0(\mathbb{Z}) \quad\text{and}\quad \bar M_{E/B}(-1) \cong \psi_f^1(\mathbb{Z}). $$ Here $(-1)$ means Tate twist i.e. tensor product with $\frac{1}{2\pi i} \mathbb{Z}$.
In particular, the local system $\psi_f^0(\mathbb{Z})$ is constant, and $\psi_f^1(\mathbb{Z})$ is just (but noncanonically) a copy of $\mathbb{Z}$ at each node.
EDIT. Some references: This goes back at least to SGA7 where nearby cycle complexes were originally defined. I highly recommend Illusie's survey Autour du theoreme de monodromie locale in Asterisque 223 (in particular, section 2.1), although it is much more general than what you need.. Later it was realized by Kato, Nakayama, and others that one can explicate in a very simple way nearby cycles in a semistable situation using logarithmic geometry (the notation $\bar M_E$ I used above was motivated by this). For this, there is another nice survey by Illusie in another Asterisque. Also see 4.9 of this.