We know that the power of the Ainur dwindles as time moves on, some like Melkor have lost power by spreading it too thinly amongst their creations, and the Balrogs and Saruman had become impotent with the deaths of their physical bodies, but did Tolkien ever speak of what would happen to them when time ends and they return to live with Eru, will this loss of power continue?
In the Ainulindalë, Tolkien writes of the Valar and Maiar (emphasis mine):
Thus it came to pass that of the Ainur some abode still with Ilúvatar beyond the confines of the World; but others, and among them many of the greatest and most fair, took the leave of Ilúvatar and descended into it. But this condition Ilúvatar made, or it is the necessity of their love, that their power should thenceforward be contained and bounded in the World, to be within it for ever, until it is complete, so that they are its life and it is theirs.
The Silmarillion I Ainulindalë
This would seem to imply that, the World having ended, the power of the Ainur would no longer be constrained by it. This also meshes with what we know about the reason for the growing impotency of the Valar; namely, that it's caused by the progression of time, something that only exists within the World:
The Valar 'fade' and become more impotent, precisely in proportion as the shape and constitution of things becomes more defined and settled. The longer the Past, the more nearly defined the Future, and the less room for important change (untrammelled action, on a physical plane, that is not destructive in purpose). The Past, once 'achieved', has become part of the 'Music in being'. Only Eru may or can alter the 'Music'.
History of Middle-earth X Morgoth's Ring Part 5: "Myths Transformed" Chapter VII "Notes on motives in the Silmarillion" (iii)
This does leave us with an interesting conundrum surrounding the "Bad" spirits, who lost their power through Wrong Action, not through the inexorable progression of Time. It would be remiss of me to not bring up the Second Doom of Mandos:
Thus spake Mandos in prophecy, when the Gods sat in judgement in Valinor, and the rumour of his words was whispered among all the Elves of the West. When the world is old and the Powers grow weary, then Morgoth, seeing that the guard sleepeth, shall come back through the Door of Night out of the Timeless Void; and he shall destroy the Sun and Moon. But Eärendel shall descend upon him as a white and searing flame and drive him from the airs. Then shall the Last Battle be gathered on the fields of Valinor. In that day Tulkas shall strive with Morgoth, and on his right hand shall be Fionwe, and on his left Turin Turambar, son of Hurin, coming from the halls of Mandos; and the black sword of Turin shall deal unto Morgoth his death and final end; and so shall the children of Hurin and all Men be avenged.
In that light the Gods will grow young again, and the Elves awake and all their dead arise, and the purpose of Ilúvatar be fulfilled concerning them.
History of Middle-earth V The Shaping of Middle-earth Part 3: "The Quenta"
This prophecy would suggest a couple of things:
- Morgoth himself will regain much of his power even before the end of the world
- The Valar won't betotally useless, since they'll be able to do battle with him
But whether we should take this as written is unclear; it's an early draft of Tolkien's which is believed to have been abandoned.
Ultimately, unfortunately, we have no basis for speculation; whatever happens to them will be up to the ineffable will of Ilúvatar.