Just for anyone who might be interested in this thread:
our daughter has recently started at MyGym in the Suning Plaza mall. It is a kind of playfull ‘toddler gym’ not really focused at gymnastics as a sport but more at overall motoric skills development and play. She is having a good time there, for the moment it is enough. http://www.mygymchina.com
At MyGym, they were very kind to also enquire for me for a real gymnastics club for children, and she gave me the following:
Chengdu Youth Amateur Sports School, Caotang East Road, Qingyang (close to Sichuan Museum and the Art&Curio market), tel. 028-87319631.
I have visited the place but did not yet see a lesson or talk to a trainer, but perhaps will do so after the summer if my daughter stays interested.