Have started Buspar (buspirone) 5mg twice daily as initial dose with Prozac 20mg that I am already taking. How soon should be the Buspar (buspirone) increased to 10mg twice daily safely without increase in side effects?

Buspar (buspirone) At that does, usually within 3 to 7 days as tolerated, unless otherwised advised by your doctor.
Depends. That is a low starting dose. I use up to 30mg twice per day. I sometimes start people at 15mg per day so if you don't have any side effects in a few days you can increase.
2 weeks. I usually recommend increasing Buspar (buspirone) to the next level in about 2 weeks depending on how well you tolerate it.