Diana's apartments were actually two separate adjoining apartments merged into one. From what I read as late as the latter part of 2006, the "main" rooms in the larger section of the apartments: the Drawing Room, the Dining Room, Diana's Sitting Room, and Charles (and later the boys') Study / Sitting Room were still empty.
The smaller apartment section (the Bedroom, Dressing Room, Wardrobe / Bath and ancillary rooms) have been remodelled into offices.
And the Princes' Nursury rooms (on the attic floor above the main apartment) are used for storage.
At present it wouldn't be possible to have these apartments turned into a Diana Museum, since they are squarely in the middle of the private residential apartments of the palace - next door to Prince and Princess Michael of Kent.
The future of those apartments - and in fact the entire private residential area of the palace - remains unknown. Given the howling of Parliament a few years back over the palace's maintenance costs, it appears doubtful that any future royals will live there, but I wonder if new courtires / staff are still being given quarters there even today.