Welcome to Ireland Reaching Out!
You have done a great job of researching your Nugent family.
I checked the subscription site Roots Ireland to see if they had any Patrick Nugent baptismal records. I found an 1806 record in Cappataggle RC (father Thomas) and an 1813 record in Tuam (father Andrew) but I don't think either record pertains to your ancestor. Many RC parishes do not have records back to the 1810 era.
What was the name of Patrick's oldest son?
Have you considered DNA testing? Maybe a descendant of John has taken a DNA test and the matching data may assist you in making the connection between Patrick and John.
Please consider adding John's very interesting story to our XO Chronicles site https://www.irelandxo.com/ireland-xo/history-and-genealogy/ancestor-data...