Your county may allow you to file a motion for temporary timeshare before your mediation is completed, but most counties don't. If not, then all you can do is wait. The fact that she agrees to paternity may speed things up a little, but paternity cases are more about timeshare (custody and visitation) and support than about fatherhood. Our court system is pathetically slow, mostly because our austerity-minded legislature won't properly fund it. Console yourself in the fact that her claim for "custody" is being severely weakened by the fact that she won't let you see your child.
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"I signed my sons birth certificate"
and "acknowledgement for paternity was signed and notarized by mom and dad"
"Her answer to my petition agrees that I'm the father and it is uncontested"
We hear this more times than we like. Unfortunately, while the term you will use in your motion is that, as a matter of law, paternity is not disputed, and or should be established, it has not been established. Signing a birth certificate obligates you to pay child support, nothing more. The acknowledgement of paternity signed by both parents is a tool to obtain a court order establishing paternity, which is the stepping stone to any timesharing. You should hire a lawyer because if she admitted in her answer that you are the legal and natural father, you should file a Motion for Entry of Partial Final Judgment Establishing Paternity, and reserving on the remaining issues. This motion can be set on the court's motion calendar (or UCD, the 8:45 a.m. calendar Monday - Thursday, for matters of less than five minutes which are non-evidentiary in nature). Once you get that order, which is easy to get, then you can file a motion or urgent motion for temporary relief, in the form of a temporary parenting plan, temporary shared parental decision-making and for holiday timesharing. Then you can attend mediation to try and broker a final resolution but you should do this immediately because if you don't, and if she decides to move out of state, there is nothing you can do to stop her before paternity is established, and as long as you do not try to tie this with timesharing, it will be easy to do, and a short-cut to establishing some temporary timesharing that is in no way tied to child support. At that point, you will be an equal parent to her in the courts, and there is no more custody in the state of Florida, just parenting plans which will designate all the things you are concerned about. Good luck.