I have 2 children. My oldest is a girl, and at the first sign she was ready to potty train, I jumped on it (right about 2 yrs. of age). Looking back I think I pushed her, and she pushed back. It was a controll issue and she was in controll no matter how hard I tried to be. She would argue, fuss and fight if I just wanted her to try and then minutes later have an accident. We continuted to have many many accidents over the next 2 years.
I had a neighbor who was brave enough to wait until her son turned 3, and he was trained in a matter of days. People judged her - I know it was hard. The reason I know it was hard is because I followed her example with my son. I started to potty train him the day after his 3rd birth day - It was a breeze!! It took 2 days and he had it down, #1 and #2! I never had to carry extra clothes, he never fought me, if I asked him to try he would try and almost always go. If he has to go he stops playing and takes care of business. My mother, mother-in-law, and most of my friends thought I was crazy to wait. I stuck to it, and I didn't let anyone pressure me into pressuring him.
My advice is wait. I know this advice can be harder to take if your granddaughter is in daycare, but if you and your daugher can wait I think you will save yourselves lots of time, energy and frustration!!
Best of luck no matter what you decide to do!
Jen mother of 2 angels. 6 year old girl, nearly 4 year old boy