J.,
I am a 44 year old woman who had almost exactly the same problem as a child. The only difference was that I was near-sighted, but I had the lazy eye. I also wore glasses from the time I was about 2 until I was about 5 for the lazy eye. The glasses had "prisms" in the center to focus my eyes to the center. At 8 years old, I began to wear glasses for the near-sightedness, and I wear contacts to this day. I had to do eye exercises (what they called pencil pushups) to get my eyes to work together, and also wear a red filter on one eye. I was told to make sure the world looked pink, which meant I was using both eyes. I also was patched for a short period of time.
OK, all this history to tell you that I am a perfectly happy, perfectly healthy adult. I was an honor student in high school and college and earned a teaching certificate as well as a psychology degree. I also have graduate level credits in psychology and am a education consultant for the local school district. You would never know that I ever had this problem, except that sometimes at night my vision gets a little doubled when I am tired. I remember telling my father after getting my first glasses that I never knew that grass looked like "ones" (blades) because I always just saw a blur. He (to this day) feels so guilty that he didn't realize it sooner. But, I don't have any problem with it! In fact, I appreciate good sight so much and I make sure that my 3 kids go to the eye doctor each year. All three (ages 17, 14, and 12) wear glasses or contacts now. No lazy eyes...just a lot of nearsightedness!
I am not an eye doctor, but I don't believe that waiting 6 weeks for glasses will hurt her irreparably. If it would, they would certainly rush them to avoid a lawsuit! If you are worried, you might want to try eye exercises, such as slowly moving your finger to touch the tip of her nose and see if she can follow your finger all the way in. Make it a game, and it may also help her focus and control her eyes.
I know how panicky you feel, but please know that you did the right thing by noticing the problem and taking action. Stay on top of it and trust your gut. This will turn out just fine...I know it!!