If by "Hamilton's Principle," you mean "the Variational Principle that Hamilton's Action is Extremized," strictly speaking, no such "Nonholonomic Constrained Action Principle" exists, except in the special case that the "nonholonomic constraints" are "integrable" --- i.e., that they are really just "disguised" holonomic constraints that have been written in an apparently non-holonomic form. (One will find a few contrary claims in a few textbooks and papers, but on careful examination, these so-called "nonholonomic variational principles" are all either ill-posed or not self-consistent unless the constraints are integrable.)
IIRC, a mathematically rigorous treatment of constrained variational problems may be found in Rund and Lovelock's
https://www.amazon.com/dp/0486658406/?tag=pfamazon01-20 but it's fairly heavy going. (Or perhaps I may be thinking of Rund's
"Hamilton-Jacobi Theory of the Calculus of Variations," which is sadly now out of print...)
There is a modification of Hamilton's Principle called the "Hamilton-Jacobi-Bellman Principle" (or just the "Bellman Principle" for short), that is used to formulate "Optimal Control Problems" --- including problems with non-holonomic constraints. However, the Bellman Principle is
not in general equivalent to Hamilton Principle, and it
does in general lead to equations of motion that are "mathematically degenerate" --- i.e., they have a nontrivial "nullspace," implying that their solutions are non-unique.
There is also Dirac's Theory of Constrained Hamilitonian Systems, but Dirac's formalism cannot be derived from a variational principle except in the special case that the constraints are integrable --- i.e., the constraints are equivalent to holonomic constraints.
