Had infections at nailbed & ingrown nail. Saw podiatrist for ingrown nail 6 mth ago nail now yellow & broken. Will it grow out? Can I use fake nail?

Nail regrowth. As long as the nailbed has not been damaged or removed, nails should regrow with time (lots of time). Sometimes the new growth is characterized by discoloration or distortion, and this is usually permanent. Occlusion of the exposed nailbed with artificial nails and chemicals may further damage the nailbed and inhibit any new growth that might occur.
Do not use fake nail. You have probably developed a fungal infection in the nail. Go back to the Dr. for treatment options. Common treatment options are topical and oral antifungals and laser treatment.
It should and yes. it can take a toenail over a year to grow out fully. Using a fake nail is fine. .