I’d like to point out that immediately after saying that you think running and gliding cannot be compared, you compare them. Running and gliding are obviously not similar, but they can be compared.
ETA: this comment may just be too snarky as well and I apologize if it is.
Humans are the fiercest predators which have ever existed on this planet. We eat sharks. I can walk into any supermarket and buy a tin of shark meat right off the shelf. We beat every single predator at its own game: we are more cunning than the tiger, more ruthless than the shark, more relentless than the wolf. Only one species has ever challenged humanity for its place atop the food chain – the sabre-tooth cat – and we grimly sought out each and every one one in every one of its dens and turned them into rugs. Not even the mighty tyrannosaurus rex was so awful in its predatory might that it threatened to drive every other species into extinction through murderous efficiency.
It was done by the Predatory Pronghorn Conspiracy! 
Thanks for all the info – I was initially going to limit the query to just predators, as per my original question, but why not expand the scope to include the speed of all sorts of animals?
By the way – what about greyhounds? I know that they are fast at the track (I am not sure how fast), but can their ability extend to rough, natural terrain very well? How would they do “out in the wild”, I wonder?
I’ll stick with the lion. Several of the “10 fastest land animals” lists have it at 4th or 5th position, with a bunch of herbivores in between it and the cheetah. These lists usually put the cape dog down at 8th or 9th. It becomes a question of how sustained a speed we are looking for. The lion isn’t going to maintain the quoted 50 mph over a very long distance, but if we’re counting short bursts, it gets the prize. The cape dog runs fast for long chases.
Bats aren’t terrifically fast flyers. They’re built for manueverability, not speed. Various sources seem to claim that the fastest bat is the Mexican Freetail, with quoted various “top speeds” between 40 and 60 mph. Far slower than fast birds, and I think we can count the fastest land animals as running faster than they fly.
I don’t think any swimming mammals will get into range, discounting instances of them riding bow waves of vessels. The fastest mammal on this guy’s list, for instance, is the killer whale, listed at 35 mph:
Since the OP acknowledges that the cheetah is first, I don’t think he’s asking about distance running.
I suspect greyhounds and whippets would do poorly in the wild because they’re so stupid. Their narrow heads have tiny little brains, and are considered among the stupidest of the domesticated dogs. They may be fast, but they’re dumb as a post. Not a desirable trait in a predator.
I’d vote Neanderthal and Cro-Magnon.
Cro-Magnon were anatomically modern humans. If you want to consider them a separate species you can’t very well say they competed with us- they became us.
Well, unless you’re the prey. 
← notice teeth
Also, I’d say it’s unfair to have purpose-bred animals compete with naturally evolved ones - predators out in the wild can’t be bred just for speed, or just for being tiny and fitting inside a purse, or whatever. It’s an unfair advantage over wild animals that have to also be able to make it out in the big bad world.
No real evidence for this. The sabre-toothed cat Smilodon (if that is what you are referring to) was restricted to the New World, and there is no direct evidence they ever interacted with humans. While it is possible that hunting by Paleoindians contributed to its extinction, the disappearance of the other megafauna that were its prey was a more likely cause.
Alaskan sled dogs get my vote for time over distance. They have two positions run and stop. And stop is only used for eating and sleeping.
There were lots of other sabre-toothed felinidae, though, not just Smilodon, and most of them existed outside the New World. Some, like Megantereon, appear to have hunted, or at least eaten, hominids.
On the other hand, I don’t know if they could really be credited as primary challengers to humans, but I doubt it.
True; however Smilodon fatalis is the species most commonly referred to as “the sabre-toothed cat.” While sabre-tooths in general no doubt occasionally took humans, considering their specialized dentition it is very unlikely that we would have been an important prey item. While it’s hard to say, since there is no modern ecological equivalent, it is generally thought that sabre-toothed dentition is an adaptation for slashing or stabbing very large prey. Leopards seem to have been a more significant predator on early humans.
Who cares?
Is there a reason to make that distinction other than it’s “unfair?” Might as well claim that lions have an unfair advantage since they evolved in an area with fast prey.
Which is your way of saying that you disqualify them from consideration because they fail to meet the criteria that is likely to define a predator?
Don’t be so sure about that. From this link:
(Bolding is mine)
Cats are getting close, though. Even lions don’t seem to be content on foot, which is forcing prey to keep pace.
It’s only a matter of time before somebody links to the YouTube video of that waterskiing squirrel.
