Bursitis is a specific
diagnosis. I think it really takes specific medical knowledge to diagnose it, so perhaps it's something else?
I'm recovering from a similar fall. Mine not only made my hip hurt, but also gave me a back spasm. I found gentle exercise, Advil, and rest worked. Yet I still have a pain in my hip when I do crossovers backward. I can make the hip pain less with a good warmup (walking) followed by stretching. Sitting too long makes it worse, so I have to get up at work and walk around. Since I've been injured a lot, these techniques have worked for me except when I have actually bruised a bone.
If you've bruised a bone the pain may last for months. There is no treatment to my knowledge other than heat and painkillers. (Yes you can bruise a bone without a bruise showing on your skin. I bruised the bones in my feet once after being thrown from a horse. There was no outward sign)
The soreness in your glutes may be due to the fact that if you injure one area of your body, another area takes up more work to overcome the weakness in the injured area.
Regardless of the actual reason for the pain after an injury, I do my best to continue to exercise, except I lower the intensity. Warmups are important, as also not letting your muscles get stiff during the working day.