I can’t claim a connection with any Oakshott’s in Cork but I do have a man with that surname marrying into an English branch of my family. He was born in Hampshire in the 1820’s and a very quick open query of FreeBMD shows lots of Oakshott vital events being recorded in the Catherington, Droxford and Portsea Island RD’s in the early decades of English civil registration. John Gresham’s surname maps only show a very few entries for Oakshott, all in Cork City. So, I would guess that your John Oakshott is English and you should try hunting in the Hampshire parish registers for him. I would also try to find occupation records especially apprenticeship records in Hampshire / England for master builders – sorry not sure where you can find these easily. Personally, I would get in touch with Hampshire Archives in Winchester and see if they can help you with possible locations of these records. It may also be worth getting in touch with the Hampshire Genealogical Society as maybe one of their membership has researched this name in this location?
It appears to be a relatively unusual surname and if you do find a connection to a James Oakshott ba. 1826/7, Durley, Hampshire marrying an Ann Hardy, I’ve researched his line forward, including New Zealand descendants and a very naughty son called Henry who is my star criminal in my family tree!