Now that the paper has been published, your role is not that of a reviewer anymore, but is the same as any reader. Anyone might read the paper and draw the same conclusions as you.
As such, you can act just as any reader would act, without having to disclose your earlier role as a reviewer. It seems to me that you then have a choice of three options:
- Write to the authors
- Write a comment to their paper
- Write to the editor, mentioning your doubts about the adequacy of references to the existing literature (including previous studies by the authors)
In your particular case, the only additional information that you (as a former reviewer) have is that the authors have been journal-shopping for their paper, and do not want to make the necessary modifications to it. This means that solution #1 is probably not going to be productive, because the authors acted in bad faith. So, you're left with #2 and #3.
If you choose to write a public comment, you definitely cannot say that you were a reviewer for a previous version of the journal. On the other hand, if you write to the editor, I think you could reasonably give him that information (“I was a reviewer of this particular paper for another journal”), because (i) it may change his point of view of the author's honesty, (ii) he will treat it as confidential information.