How quickly does a UTI spread from the bladder to the kidneys? I think I've had one for a few days but my doctor can't fit me in for another two days.

Not that quickly. Not all bladder infections spread to the kidneys. When they do, it takes more than a few days, especially if you simply keep yourself well hydrated, drinking plenty of water. A good way of assuring adequate hydration is drinking a glass of water every time you void until your urine is very faintly yellow.