No, donors have no rights to the child born as a result of their donation. The donor is not the legal parent, has no legal obligation to the child or children, will not be named on the birth certificate and will not have to pay financial support to the child.
Conversely, when fresh donor sperm is used outside of an HFEA licensed clinic, the donor will be considered by law the child's legal father, with all the rights, responsibilities and obligations it involves.