USLEEP is not required but highly recommended due to how many weapon-related bugs it fixes.  Additionally, USLEEP enhances the dual-wielding power-attack speed of daggers, which fits nicely with this mod.

All standard two-handed weapons, war axes, and maces are now faster both in terms of total speed and in relation to one another.  Daggers and one-handed swords have been included below for comparison:

                             Vanilla                             New
Daggers                 1.3                                  1.3
Swords                  1                                      1
War Axes               0.9                                  0.975
Maces                    0.8                                  0.95
Greatswords         0.7 (USLEEP: 0.75)       0.85
Battleaxes             0.7                                  0.825
Warhammers        0.6                                  0.8

The Longhammer is still as fast as a mace, now at 0.95.

The Woodsman's Friend is now as fast as a standard battleaxe at 0.825.

In vanilla the Ebony Blade was as fast as a one-handed sword at 1.  USLEEP made it as fast as a standard greatsword at 0.75.  Since it looks like a light weapon and can't be tempered, I think setting the speed between that of greatswords and maces at 0.9 is a fair compromise.  I've also adjusted the stagger from 0.75 (USLEEP missed this) to between that of greatswords and maces at 1.05.  This has nothing to do with weapon speed, but I just couldn't leave it broken like that.  The same goes for the next entry on this list.

The Headsman's Axe only had a stagger of 0.85 (USLEEP missed this too), so I adjusted it to the battleaxe standard at 1.15.

In vanilla Volendrung was as fast as standard greatswords and battleaxes at 0.7.  USLEEP left its speed at 0.7 while updating greatswords to 0.75.  I don't know why it was ever faster than standard warhammers -- there is no lore-based or logical explanation that I know of -- but for the sake of consistency it is now as fast as standard battleaxes at 0.825

The Rueful Axe is said to be "an incredibly powerful axe."  However, it does the same damage as a glass battleaxe (compare this to Wuuthrad, which is on par with a daedric battleaxe), only has a speed of 0.5 (instead of 0.7), and has an enchantment which is not particularly useful (20pts stamina damage).  Now, maybe Clavicus Vile, being the shady dealer he is, has tricked the player character into accepting a low-grade artifact.  However, this artifact was supposedly used by a dude in a robe to kill a werewolf, so it shouldn't be this bad of a weapon.  I also think there should be some incentive for killing Barbas in order to keep it.  After all, you can't swindle someone into accepting a crappy weapon if you've also given them a chance to test it out.  As such I think the Rueful Axe should at least be as fast as a standard battleaxe.

Test Vorpal Sword has received the same adjustments as the Ebony blade, if you want to mess around with a god item that was left in by the developers.

Ash spawn weapons and most other unplayable weapons have had their speeds adjusted accordingly.  Using other mods to make them playable will not change this (unless those mods adjust their speed specifically).

Woodcutter's axes, pickaxes, and giants' clubs have been left unchanged.

Mod weapons will obviously remain at whatever speed they were originally set at.  I might make patches for a couple of weapon mods at some point.