If Scar was the rightful king, I expect his reigning parent would have trained him to become a good king throughout his life, the way we saw Mufasa start to do with Simba (and undoubtedly would've done if he hadn't died suddenly). I expect Mufasa went under the same kind of training before his reign. Whether or not Scar did too is unknown; I think he should have done since it was unknown to them that Mufasa would definitely have a cub, but if he was trained them his training may not have been as serious (or maybe Mufasa was expected to continue his training once he was king).
Should this be the case, then I'd like to think Scar would have turned out to be a somewhat better king (even if he did still end up less responsible than Mufasa as a king). I also had a feeling that, in the film, it was the hyenas who caused the lionesses to overhunt, which was how the Pridelands became destroyed? I'll have to check that information, but if that's the case I think that would change too...
If Scar was king then I don't think he would actually have an army of hyenas. I don't know how he first befriended them in the film, but thought it might have something to do with the fact that he had a need for power and the rest of the pride followed his brother, not him. That may not be true though; perhaps there was another reason why he would hang out with them and being king might not have stopped this. But basically, if he had a pride of lionesses then he may have not bothered with hyenas too.
Whether or not the Pridelands ended up getting destroyed, I would still like to think that Scar was naturally very power-hungry. Sure, maybe it's normal for princes to feel a bit jealous when their brother's the king, and sure, he would've been disappointed when Simba's birth meant he was no longer next in line. But even so, to go as far as killing Mufasa and banishing Simba suggests that there's definitely something not right about him!
I don't think Mufasa would've been like that. I imagine he would have been more supportive of Scar during his reign. Maybe he too would have felt a bit of jealousy if Scar had a cub, but even if he did then certainly not enough to commit a crime.
During Scar's reign (in the film) he was shown to just enjoy the power and glory, and didn't care about his responsibilities. So if he had a cub I doubt he would have done a very good job training them up. Of course, like I said earlier if Scar himself received the right training he may have been a better king anyway, so his cub's training may depend on his own training.
In this alternate version, I would still like to think that Simba's parents are still Mufasa and Sarabi. But I also think that, as a king, Scar would have been likely to end up with a mate and cubs too. Somehow, I think it would've been easier for him to find a mate if he'd been king all along!
This would have made Simba the cousin to the heir of the throne. In the canon world, he acted a bit like the lion version of a typical "rich kid" in places. Still part of the royal family, he may be a little like this in the alternate universe too, though not quite as much. Scar's cub, however, I imagine to be very snobby and irresponsible, taking their luxuries for granted.
And perhaps Mufasa would have stepped in and attempted to teach Scar's cub how to be a good monarch. Perhaps eventually this cub would change, start to look up to Mufasa instead of Scar, and Scar would start to feel jealous, thinking that Mufasa is trying to "steal" his cub away from him.
Or it could go in another direction. Maybe Scar trains this cub to be a fighter and how to take control of the pride. Perhaps eventually, when both Mufasa and Scar are old and Scar decides to resign as king, the pride get into a fight, debating that Simba should be the new king, instead of letting Scar's child take the throne. The two candidates could battle each other for the throne.
Or maybe another direction: during Scar's reign, Mufasa decides that Scar fails at being king, takes Sarabi and Simba and start off their own pride. Except I think it would probably be out of character for Mufasa to abandon Scar like that, and would hope he'd be more likely to help him and his cub than this!
Well, that's a few of my ideas.
It is an interesting topic though. I like "alternate universe" theories. ;D