This disconnection must have had something to do with the Timothy Thadius problem.
Clearly, I know that Timothy was Robert Emmett's father, but I don't believe that Thadius was his name.
I am not aware of any document that Timothy signed his name as anything other than Timothy. Since he was born in Ireland and a Roman Catholic, his baptismal record IN LATIN may well have been Thaddeus Lynskey. His mother would have been Maria Mullen and his father would have been Joannus Lynskey. That's because the Church records were written in Latin. Thaddeus is the Latin form of Timothy.
Although John had no Irish brogue, Hannah had a heavy brogue, my father told me that she always called him Thady, a nick name for Thaddeus.
If Hannah provided the information, she referred to him as Thady. If he provided the information his name was Timothy. I don't think that very many people give their child a baptismal name that is a Latin equivalent of it's given name!