*Due to rules of your state bar or mine, nothing herein is intended as legal advice, only intended as general information to better help yourself.
Actually, it may be possible for the 'possibly real' dad to come forward and file a petition with the Court asking for a paternity test based on his testimony/affidavit that you two were intimate at a time when science says the child was likely conceived and that he now has reason to believe that he is the father. He can use an affidavit by you where you state under oath that you had relations on or around XX date, and that you learned you were X weeks pregnant on or around YY date, which was X number of days after you started having relations with Timmy. Once he is granted the right of a paternity test he can prove parentage and seek custody and/or visitation. It may NOT negate Timmy's right to visitation or even custody if he fights for it and the court finds it in the child's best interest, particularly if they find Timmy to be "in loco parentis" to the child - the only father the child has known. But, it can get the other dad some rights. As for your rights, you need to make those motions to enforce the child's right (and yours) to a relationship with you - your visitation rights, etc.
I hope this helps clarify for you.Because I help people like you here, for a living---this is not a hobby for me---I sincerely XXXXX XXXXX abiding by the honor system with regard to Accepting answers, by Clicking your ACCEPT button now. Feel free to follow up after, if you need clarification. An ACCEPT also assures that I can assist you again. A BONUS is a wonderful way to tell the expert her time and effort are appreciated. I wish you the best in your future.
DISCLAIMER: Answers from Experts on JustAnswer are not substitutes for the advice of an attorney. JustAnswer is a public forum and questions and responses are not private or confidential or protected by the attorney-client privilege. The Expert above is not your attorney, and the response above is not legal advice. You should not read this response to propose specific action or address specific circumstances, but only to give you a sense of general principles of law that might affect the situation you describe. Application of these general principles to particular circumstances must be done by a lawyer who has spoken with you in confidence, learned all relevant information, and explored various options. Before acting on these general principles, you should hire a lawyer licensed to practice law in the jurisdiction to which your question pertains.
The responses above are from individual Experts, not JustAnswer. The site and services are provided “as is”. To view the verified credential of an Expert, click on the “Verified” symbol in the Expert’s profile. This site is not for emergency questions which should be directed immediately by telephone or in-person to qualified professionals. Please carefully read the Terms of Service (last updated February 8, 2012).