  The Torsion Anomalous Conjecture states that an irreducible variety $V$
embedded in a semi-abelian variety contains only finitely many maximal
$V$-torsion anomalous varieties. In this paper we consider an irreducible
variety embedded in a product of elliptic curves. Our main result provides a
totally explicit bound for the N\'eron-Tate height of all maximal $V$-torsion
anomalous points of relative codimension one, in the non CM case, and an
analogous effective result in the CM case. As an application, we obtain the
finiteness of such points. In addition, we deduce some new explicit results in
the context of the effective Mordell-Lang Conjecture; in particular we bound
the N\'eron-Tate height of the rational points of an explicit family of curves
of increasing genus.
