There is a VERY large women’s and children’s hospital to the west of the no 3 ring road that I gather is generally considered the place to go by the Chinese (my stepdaughter went there for a C-section and had no complaints – 2 patients to a room). It’s the no 1 maternity hospital for Chengdu so shouldn’t be difficult to locate.
I would tread with caution as far as “private” hospitals are concerned as they may be more motivated by the potential profit and “international” hospitals (clinics) may not be blessed with the same levels of expertise and equipment that is available in the 1st tier hospitals who charge a fraction of the price. While you might find that the smaller hospitals are less crowded, in my experience English speaking Chinese doctors are not uncommon.