The text of the Second Prophecy, as given in the 1937 Silmarillion, indicates that Morgoth will not participate:
...and the black sword of Turin shall deal unto Morgoth his death and final end...
Use of the phrase "final end" here tells us that Morgoth will cease to exist, and since this was to happen before the Second Music, it's not possible for Morgoth to participate.
This passage was unchanged in the 1950s revisions, so it represents the latest concept of the Dagor Dagorath.
Removal of the Second Prophecy of Mandos from the published Silmarillion was made on the basis of the closing lines of the Valaquenta, which CT re-purposed as his closing lines for the Quenta Silmarillion:
Here ends The Valaquenta. If it has passed from the high and beautiful to darkness and ruin, that was of old the fate of Arda Marred; and if any change shall come and the Marring be amended, Manwe and Varda may know; but they have not revealed it, and it is not declared in the dooms of Mandos.
(My emphasis, sourced from the Valaquenta text in HoME 10.)
As CT notes, this means that:
The Second Prophecy of Mandos has now therefore definitively disappeared.
...obviously, because if an Unmarring is "not declared in the dooms of Mandos" there can be no Second Prophecy.
There's nothing I'm aware of to indicate the ultimate fate of any of the other evil beings.