A small prefaction first: when a hunter is asleep or stunned, any hit will free them of that status; if a hunter is paralyzed, it takes a hit that sends the hunter flying, either horizontally or vertically. Since the kick doesn't send hunters flying unless when having the Felyne Kickboxer skill, it wasn't helpful against paralysis.
The special kick attack is gone from Monster Hunter World, and is now it's only part of the Greatsword's moveset. You can guard and press the Triangle/Y button to perform a kick.
Other than that, the same rules apply: in order to break an ally out of sleep and stun you only need to hit them with any weapon attack. If you want to save a hunter from paralysis, however, you'll need to use specific moves, such as:
- The Greatsword's rising slash
- The Sword and Shield's second or third hit of the shield bash combo
- The Hammer's superpound
- The Hunting Horn's superpoundcharged uppercut
- The Gunlance's wyvern fire
- The Lance's charge
- The Switch Axe's rising slash in axe mode
- The Charge Blade's super amped elemental discharge
- Certain Bowgun ammo such as Wyvern Ammo, Sticky Ammo, and Cluster Ammo
I don't believe any weapon other than the Long Sword, Dual Blades, Insect Glaive, and Bowones mentioned can help a paralyzed ally with any move.