There are two main classes of solutions to be discussed in this regard.
"Sufficiently" Smooth Solutions
In Strang's classical paper it is shown that the Lax equivalence theorem (i.e., the idea that consistency plus stability implies convergence) extends to nonlinear PDE solutions if they have a certain number of continuous derivatives. Note that that paper is focused on hyperbolic problems, but the result carries over to parabolic problems. The number of derivatives needed is a technical point, but this approach is usually applicable to solutions that satisfy the PDE in a strong sense.
At the other extreme, we have PDE "solutions" with discontinuities, which typically arise from nonlinear hyperbolic conservation laws. In this situation, of course, the solution cannot be said to satisfy the PDE in the strong sense, as it is non-differentiable at one or more points. Instead, a notion of weak solution must be introduced, which essentially amounts to requiring that the solution satisfy an integral conservation law.
Proving convergence of a sequence of solutions is also more difficult in this case, as $L_p$-stability is not enough; usually the sequence must be shown to lie in a compact space, such as the set of $L_\infty$ functions with some finite maximum total variation.
If the sequence can be shown to converge to something, and if the method is conservative, then the Lax-Wendroff theorem guarantees that it will converge to a weak solution of the conservation law. However, such solutions are not unique. Determining which weak solution is "correct" requires information that is not contained in the hyperbolic PDE. Generally, hyperbolic PDEs are obtained by neglecting parabolic terms in a continuum model, and the correct weak solution can depend on exactly what parabolic terms were discarded (this last point is the focus of the paper linked to in the question above).
This is a rich and involved topic, and the mathematical theory is far from complete. Most convergence proofs are for 1D problems and rely on specialized techniques. Thus nearly all of the actual computational solutions of hyperbolic conservation laws in practice cannot be proved convergent with existing tools. For a practical discussion from a computational standpoint, see LeVeque's book (chapters 8, 12, and 15); for a more rigorous and detailed treatment I'd suggest Dafermos.