Vrftsjtryk

I think it might be a tossup between a dragon and a phoenix but I would like it if there was something in canon about this.

Ignoring those creatures that are able to simply appararate from place to place instantaneously (including the Phoenix, the Diricawl, the Elf and the Wizard), the fastest seen is probably the Zouwo.

The Zouwu is a monstrously large feline beast—as big as an elephant—with a striped body, scraggly mane, four fangs that curl up out of its mouth, and long sharp claws. Perhaps its most distinctive feature is its disproportionately long and ruffled multicolored tail. Native to China, Zouwus are incredibly powerful and fast, capable of traveling 1,000 miles in a day. The Zouwu in Fantastic Beasts: The Crimes of Grindelwald was captured by Skender, the owner and Ringmaster of the wizarding world’s Circus Arcanus, and bears the scars of its abuse.

_{^{Beware! New Beasts Ahead in “Crimes of Grindelwald”}}

Other contenders mentioned in the original Fantastic Beasts book include the Nogtail (albeit presumably over a much shorter distance)

Nogtails are demons found in rural areas right across Europe, Russia
and America. They resemble stunted piglets with long legs, thick,
stubby tails and narrow black eyes. The Nogtail will creep into a sty
and suckle an ordinary sow alongside her own young. The longer the
Nogtail is left undetected and the bigger it grows, the longer the
blight on the farm into which it has entered. The Nogtail is
exceptionally fast and difficult to catch, though if chased beyond the
boundaries of a farm by a pure white dog, it will never return. The
Department for the Regulation and Control of Magical Creatures (Pest
Sub-Division) keeps a dozen albino bloodhounds for this purpose.

The Snidget (on which the Golden Snitch is based)

The Golden Snidget is an extremely rare, protected species of bird.
Completely round, with a very long, thin beak and glistening,
jewel-like red eyes, the Golden Snidget is an extremely fast flier
that can change direction with uncanny speed and skill, owing to the
rotational joints of its wings.

and the Winged Horse (of which there are multiple varieties including the Thestral)

Winged horses exist worldwide. There are many different breeds,
including the Abraxan (immensely powerful giant palominos), the
Aethonan (chestnut, popular in Britain and Ireland), the Granian (grey
and particularly fast) and the rare Thestral (black, possessed of the
power of invisibility and considered unlucky by many wizards). As with
the Hippogriff, the owner of a winged horse is required to perform a
Disillusionment Charm upon it at regular intervals.

And presumably the usual run of flying beasties like the Hippogriff and various Dragons although their speed is apparently not worthy of mention.