Well, I myself can't give input for the other verses in question here, but I happen to know a LOT about the Lovecraft/Cthulhu Mythos verse, so here's my two-cents on the Outer Gods (i.e. the Tier 1-A's and above in the Cthulhu Mythos) having True Godly regen:
As someone previously mentioned, the Outer Gods all have existences which are sustained by Azathoth's dream. By virtue of this alone, they should have True Godly regen. Although, I will grant that a counter-argument could certainly be made to this as well, as it can be hard to exactly interpret what the true nature of the Outer Gods is really supposed to be.
SOME of the Outer Gods, however, most DEFINITELY have True Godly. One I can think of, in particular, is Yog Sothoth. Yog Sothoth is the Gate of the Silver Key as well as the Key and the Gatekeeper, and is coterminous with all of physical and metaphysical reality while simultaneously existing outside of any dimensions of time and space. Essentially, Yog Sothoth's existence is a fundamental constant of the entire Lovecraft Multiverse, as nothing can even exist without Yog Sothoth existing, and everything that exists is merely a facet of Yog Sothoth, so to speak. It would pretty much be impossible to permanently destroy Yog Sothoth without destroying literally the entirety of the (high-outerversal-level) Lovecraft Multiverse aside from Azathoth itself, which would only be possible either for a Tier 0 entity (like, I guess Kami Tenchi could do it, or something), a "stronger" Tier High 1-A entity (Featherine Augustus Aurora is one who comes to mind as possibly this powerful) than Yog Sothoth, or by causing Azathoth to wake up. So I think we don't really need to question wheteher Yog Sothoth, for one, has True-Godly regen; it definitely does.
Not so sure about the other Outer Gods. Whether they too count as "fundamental constants of the multiverse" is very much up for debate.