In the US it's just plain Gymnastics.
http://www.insidegymnastics.com/
http://en.wikipedia.org/wiki/Gymnastics
http://www.usa-gymnastics.org/
http://www.ncaasports.com/gymnastics/womens
You can refer to Women's Gymnastics/Men's Gymnastics

See this example (there are millions):
"Behind the Team" takes fans to the USA Gymnastics
Women’s National Training Center

June 27, 2007

INDIANAPOLIS, Ind., June 27, 2007 — "USA Gymnastics: Behind the Team" takes fans and sports enthusiasts inside the U.S. Women’s National Team Training Center located at Bela and Martha Karolyi’s Ranch in Huntsville, Texas, in the newest episode of the online series available only on the AT&T blue room Team USA site (www.attblueroom.com/teamusa). Fans not only see the gymnasts training, but also have a glimpse into their life at the ranch outside of the gym, from meals to hanging out in their rooms to the resident wildlife.

Bela Karolyi shares how the Women’s National Training Center became a part of the legendary Karolyi Ranch, and national team members talk about training and life at the Ranch and the role it plays in developing friendships and the world’s best women’s gymnastics team.


Just found the official site for the British team and it's also "gymnastics" or "artistic gymnastics":
http://www.british-gymnastics.org/