This may help,
Landed Estates, NUI Galway
Go to Browse Go to HOUSES under H you will see 2 x Hampstead listed, 2nd one down this is your Hampstead House.
I have checked the BDM records and none give the address as Hampstead House or The Elms.
This John H Bainbridge moves to another area of Cork by 1858. He was in France before he was in Cork.