In the somewhat canon novel "Over Heaven" there are subtle hints that The World doesn't really belong to DIO but rather to Jonathan. I personally liked this theory a lot and I want to elaborate on it for a bit more.
My first argument to this would be the fact that Stands can be inherited. Due to this, it would be healthy to assume that Jotaro inherited the power of time stop from Jonathan (it skipped generations, just like Hamon) as I see absolutely no reason for DIO and Jotaro to have the exact same power.
Furthermore, there is Joseph and Hermit Purple. Araki said that Hermit Purple is not actually a true Stand, but rather a Stand-like manifestation of one's Hamon, which means that every Hamon user would have it as their "Stand". Now, we know that because Hamon can linger in a body even after death, Jonathan's body still contained some of it, which prevented DIO from fully merging with Jonathan's body. However, DIO also possesses a Hermit Purple, which meant that Jonathan's Hamon was present in his body until he took Joseph's blood (which I believe he did to cancel out Jonathan's lingering Hamon by mixing it with Joseph's and bringing it to zero). Now, my theory is that Joseph never actually awakened his Stand, and that it was only half manifested as Hermit Purple. We saw on the example of Polnareff and Silver Chariot, that the appearance of a Stand can be altered if it's user is dismembered. Furthermore, Joseph was never hit with the Arrow, so like Star Platinum, his Stand was awakened as a sort of echo from The World's awakening, and didn't reach full potential (note that Jotaro figured out time stop only near the very end of the Part 3). To summarize it all up, I believe that Hermit Purpleˇ#2 isn't really Jonathan's Stand, but rather manifestation of his Hamon and the extension of The World.
(The section above is messy as hell, I know. Sorry for that.)
Thirdly, it has been said that DIO only managed to keep The World under control due to his elite charisma, and that if anyone else were to have it, it would run out of control. Remember how Star Platinum rejected Jolyne because they weren't compatible; I think that The World would do the same. Now, DIO should have no trouble controlling The World, because that is the extension of his own strong spirit. Or is it? What if it was the extension of Jonathan's spirit, which is why DIO had to actively keep it under control? That would also explain the Hermit Purple#2 and why was DIO's time stop limited to 5 seconds (like a normal human) before fully merging with Jonathan's body, even though his stamina was already unlimited.
It is healthy to assume that Enya shot DIO with the Arrow somewhere on his body: the body that still belonged to Jonathan, and would thus awaken his Stand, rather than DIO's. It is shown that Stands can linger after death (Weather Report, Notorious B.I.G, Limp Bizkit, Sticky Fingers, etc.) which leaves the possibility of The World lingering in Jonathan's body.
It's also possible that the undead cannot awake their Stands at all. The only undead Stand user other then DIO, Vanilla Ice, had his Stand before DIO turned him to vampire.
In it's present state, this theory is still a bit (read: a lot) rough around the edges. Due to this I would appreciate if you approached it with care and not instant and in-constructive criticism. Just be calm, cool and polite and we can discuss it like proper gentlemen and ladies.