Be warned that even in single-threaded code, replacing a ConcurrentHashMap with a HashMap may not be safe. ConcurrentHashMap forbids null as a key or value. HashMap does not forbid them (don't ask).
So in the unlikely situation that your existing code might add a null to the collection during setup (presumably in a failure case of some kind), replacing the collection as described will change the functional behaviour.
That said, provided you do nothing else concurrent reads from a HashMap are safe.
[Edit: by "concurrent reads", I mean that there are not also concurrent modifications.
Other answers explain how to ensure this. One way is to make the map immutable, but it's not necessary. For example, the JSR133 memory model explicitly defines starting a thread to be a synchronised action, meaning that changes made in thread A before it starts thread B are visible in thread B.
My intent is not to contradict those more detailed answers about the Java Memory Model. This answer is intended to point out that even aside from concurrency issues, there is at least one API difference between ConcurrentHashMap and HashMap, which could scupper even a single-threaded program which replaced one with the other.]