No, there is no way to increase swing speed. It was thought that raising Adaptability increased swing speed but this theory has been thoroughly dis-proven.
NOTE: expending your Stamina through attacking actually puts it into a negative value, increasing the recovery time.
A few tips to help you with your stamina management:
- The lower your equip % the faster your stamina recovers
- Try raising your Endurance to get more stamina to begin with
- Eating Green Grass will increase your stamina regen along with the Cloranthy Ring but they do not stack
- The Cloranthy Ring does however stack with this shield
It is also possible to change your attack direction mid-swing with greatswords/ultra-greatsword with the Left-stick