Let $X_t$ be a family of complex manifolds of dimension $n$ over a punctured disc $D^\circ=\{x \in \mathbb{C} \mid 0 < |x| < \epsilon\}$ and assume that we have chosen a model $\mathcal{X}_t$ over the disc $D=\{x \in \mathbb{C} \mid |x| < \epsilon\}$ such that $\mathcal{X}_0$ has finitely many nodes $x_1, \ldots, x_m \in \mathcal{X}_0$ as singularities. The monodromy operator $T: H^n(X_t) \to H^n(X_t)$ is given by the Picard-Lefschetz formula in terms of the vanishing cycles $\delta_i$ of the nodes $x_i$: $$ Tx = x - (-1)^{n(n-1)/2}\sum_{i=1}^n \langle x, \delta_i \rangle $$ but it clearly does not depend on a particular model $\mathcal{X}$. If the central fibre $\mathcal{X}_0$ is modified, for example, by blowing up a node, then the vanishing cycles of the nodes in the central fibre of the new model must be such that the above formula still gives the same transformation.
I am trying to see what happens in the example of the degenerating family of elliptic curves $\mathcal{E}_t: y^2 = x(x-1)(x-t)$. Here the central fibre is a nodal cubic and the corresponding vanishing cycle is one of the generating circles of the torus which becomes pinched in $\mathcal{X}_0$ and turns into a node. Let $\mathcal{E}'$ be the family obtained after blowing up the node, then $\mathcal{E}'_0$ has two nodes $x'_1, x'_2$. What are the vanishing cycles of $x'_1, x'_2$? What happens after I blow up $x'_1$ or after I blow up a smooth point of $\mathcal{E}'_0$, what will be the vanishing cycles of the new nodes?