So sorry to hear about your mother. The GOOD news is that her urologist is doing everything right! A good urologist does go back in about six weeks to repeat the TURB (biopsy). This way he/she is sure that enough of the tissue in the area where the cancer was found is taken to be sure of the stage/grade.
The stage, in your mom's case 1, indicates whether it has spread or not. The stage 1 says that it is still localized on the surface, and has probably not spread into the muscle. The next biopsy will confirm this. Grade, in this case high, indicates how rapidly the cells are dividing. A high grade tumor is much more likely to spread rather rapidly.
The treatment that your mother's doctor is recommending is "spot-on" for her diagnosis. After the diagnosis is confirmed and refined by a second biopsy, a course of BCG is indicated for high grade bladder cancer. Many of us here on the Forum have had it. For example, I was diagnosed 7 years ago with high grade, went through a complete series of BCG, and have been cancer-free ever since.
Oncologists are not usually involved in bladder cancer treatment unless it has spread or cannot be treated with agents such as BCG.
Your mother is very lucky to have you by her side.
Sara Anne