Leo Oppenheim's translation of the Nabonidus Chronicle can be found in J. B. Pritchard (ed.) Ancient Near Eastern Texts Relating to the Old Testament (= ANET; 1950, 1955, 1969).
The standard edition is A.K. Grayson, Assyrian and Babylonian Chronicles (= ABC; 1975; ISBN978-1575060491)
A translation of Chronicle 25, discovered after the publication of ABC, was published by C.B.F. Walker "Babylonian Chronicle 25: A Chronicle of the Kassite and Isin Dynasties", in G. van Driel e.a. (eds.): Zikir Šumim: Assyriological Studies Presented to F.R. Kraus on the Occasion of His Seventieth Birthday (= Fs. Kraus; 1982).
John Brinkman revises Grayson's reading of ABC 1 in: "The Babylonian Chronicle revisited" in T. Abusch, J. Huehnergard, P. Steinkeller (eds.): Lingering over words. Studies in ancient Near Eastern literature in honor of William L. Moran (1990 Atlanta; ISBN978-1555405021)
Fragments of the chronicles that are relevant to the study of the Bible, can be found in William W. Hallo (ed.), The Context of Scripture, volume 1 (2003 Leiden and Boston; ISBN978-9004106185). This book also contains the Weidner Chronicle.
A recent update of ABC is Jean-Jacques Glassner, Mesopotamian Chronicles (= CM; 2004, ISBN978-1589830905; French version 1993, ISBN978-2251339184)
An even more recent update of ABC is Amélie Kuhrt, The Persian Empire: A Corpus of Sources of the Achaemenid Period (Routledge, 2007; ISBN978-0415552790)
The publication of I. Finkel & R. J. van der Spek, Babylonian Chronicles of the Hellenistic Period (= BCHP) has been announced.