I agree with Cadet, keep it slow. Let it develop organically, as it will.
Give what you want to give and receive what she is willing to offer. Nothing more, nothing less, with no expectations.
Regarding the children, you would never and should never aspire to replace their father. If / when you do meet them, you are their mother's boyfriend. At best for them, you are their friend and "an adult" in their life. Be that, responsibly. All parental decisions should be appropriately relegated to their mom (or dad; if relevant).
Do what you can that will create awesome memories which will neither define whom you are and which will stand on their own, regardless of what ever may happen in the future between the two of you.
