It takes very little to obtain Turing completeness. For example the While programming language and Counter Machines are Turing complete and are easily recognized as a subset (up to syntax) of all three languages.
The While language is also good for an intuition of Turing completeness. You need
Potentially infinite loops, because Turing complete computations may not terminate.
Condition-dependend behaviour, so that you can break out of loops (important: in contrast to
for loops, the time after which the loop ends is not be known in advance).
A data type with an infinite domain.
You promised not to nitpick, but if you did one would have to either
replace your languages by "conceptual" variants thereof where integers are unbounded, or
implement unbounded integers in a library (like GNU MP). When you do that, you inevitably use the "true" third ingredient of your languages: They achieve infinite-domain data through dynamic allocation.
All of the above would as well hold for C. Yet your question, at least the title, is specifically about OOP languages. Basically, your three languages are extensions of C by objects. My Turing-complete subsets then discard these extensions. At the moment I cannot come up with natural subsets of any of these languages where some OOP feature is essential for Turing completeness.
As for simple, natural, polynomial time reductions to Turing machines: Polynomial time is possible, natural in a sense, simple I do not think (such a reduction is a full-feature compiler for the language, so it cannot be any simpler than the language is).
Let there be three tapes. One for the heap, one for the stack and one for indexing. The heap and the stack would contain sequences of substrings like
 which encode unbounded integers (83 in this case). When used as pointers, these integers are somehow mapped to positions in the heap or the stack. Indexing an element of the heap or the stack works by copying the pointer to the indexing tape and then moving to the respective part of the tape by using the indexing tape as a counter.