Wilkins' "prediction" was more an educated guess, based on his experience with his own wife than anything mystical. A common theme with any story dealing with immortality (primarily "Highlander," and "Amsterdam" in the movies and TV, plus all of the other vampire themed shows) is the tragedy of one partner remaining eternally young while the other grows old and dies. That's what he was talking about, and why Buffy could never have anything vaguely resembling a normal life and relationship with anyone, including either Angel or Spike who are immortal.
Likewise, with magic no longer a valid force in the Buffyverse, any hope of Angel winning the Shanshu is pretty much invalidated especially considering the fact that the award was supposed to come from the SENIOR PARTNERS for his help in an apocalypse, and the Senior Partners left Earth in the same miniseries that won Spike the captaincy of his space ship.
And Spike, who did the unbelievable in regaining his own soul, MIGHT consider doing the unthinkable and attempt to regain his mortality, but, once again, that "no more magic" thing kicks in
Of course, what Wilkins either didn't consider or just didn't mention is that, given that Buffy, Angel, or Spike, or any combination of the three, might very well die tomorrow or even late this afternoon, the whole mortality/immortality issue isn't as immediately valid as it might otherwise be.
Then again, maybe he was being absolutely truthful, especially if he was seeing Cordy, as a Higher Being, and therefore immortal, in Angel's future.