First of all, if a stack of $M$ branes is rotated relatively to a previously coincident stack of $N$ branes, it's clear that the degrees of freedom that encode the relative angle $\theta$ are nothing else than the transverse scalars determining the position/orientation of these two stacks. Any D-brane or any stack of D-branes may be rotated in any way and the quanta of the scalar fields that remember the positions are just open string modes attached to these D-branes with both endpoints. If you study the location/orientation of a D-brane or a stack of D-brane, it's its own degree of freedom that has nothing to do with the behavior of other D-branes.
So the Higgs fields arise from the normal scalars determining the transverse positions of these stacks of D-branes and the action for these D-branes is still the same D-brane Dirac-Born-Infeld action.
Now, you apparently want to see how this degree of freedom that you call $\theta$ – it's just an awkwardly chosen "degree of freedom" that can't be invariantly separated from other degrees of freedom determining the shape of the D-branes – break the $U(M+N)$ symmetry down to $U(M)\times U(N)$ and your wording makes it sound like you believe it is just ordinary Higgs mechanism in the whole space.
However, it's important to realize that this breaking of the gauge symmetry doesn't occur uniformly in the whole space. In fact, near the intersection of the stacks, the gauge symmetry is approximately enhanced to the original $U(M+N)$. What's important is that the off-block-diagonal blocks transforming as $(M,N)$ under $U(M)\times U(N)$ arise from open strings whose one end point sits at one stack and the other end point sits at the other stack. The distances between the stacks go like $\theta \cdot D$ where $D$ is the distance between the intersection. So the open string modes get an extra mass $\theta D T$ where $T$ is the string tension.
To summarize, the symmetry breaking is always described by the ordinary (non-Abelian) Dirac-Born-Infeld action. Your "rotation of stacks relatively to each other" only differs from the ordinary "separation of parallel stacks in the transverse dimension" by the fact that the distance/separation between the stacks depends on the location along the branes. It is meaningless to ask for any new actions because the (non-Abelian) Dirac-Born-Infeld action always describes all the low-energy dynamics of similar systems. The stringy/D-brane dynamics is always governed by the same laws and one should only learn it once.
Let me mention that whenever the distance between the stacks exceeds the string length, all the off-diagonal open string modes are string-scale heavy and it is inconsistent to keep them unless you also keep the excited string harmonics in the spectrum. So a derivation of "effective field theory" that would neglect the stringy tower but that would just freely describe the higgsed large string-scale masses of the off-block-diagonal modes would be inconsistent.