Hello Edenhope,
You have your work cut out for you - there are a lot of William Gleesons from the time/place you are searching!
I have no information on your direct ancestor, but I do have a few thoughts. If your William Gleeson's eldest son was named Edward, the odds are great that his father was named Edward. I'd begin with that premise as Edward is slightly less common than William. Also, the spelling was generally Gleeson rather than Gleason in those days.
I happen to have a Timothy Gleeson line on my tree that probably connects to yours. My earliest Timothy Gleeson was born about 1800 in Borrisokane - quite near Nenagh. He married Margaret Burns and they emigrated to Nepean, Ontario, Canada about 1824. Their children were Anne, John, James, Ellen, Viena, Patrick, Timothy, Catherine, Cornelius, Michael, Bridget, Margaret.
I haven't traced the line back any further than Timothy, but from his eldest son's name, I'm guessing his father was likely named John and his mother, Anne. If I were working your line, I might try researching a northern Tiipperary family with John and Anne Gleeson as the parents and a son named Edward (and another named Timothy and possibly, a William). I'd begin in and around Nenagh, Borrisokane, Terryglass, etc and search the NLI registers - page by page for the pertinent years. Conversely, you can register with RootsIreland for as little as one day (fee-based) and see if they've indexed records that will lead you to the family.
It's a lot of time/effort but I've broken a number of brick walls in exactly that way. I wish you well.
Frances