Be very careful not to order any certificates for birth registrations outside of the London area. There are a couple of families living and having children at the same time and same names as your Joseph and George. These can be seen on the census pages and are not your family.
As far as I can see, there are no registrations for your Joseph b 1848 and George b 1850. Neither are there any registrations for the same time period where the children are just registered as Ludgate Male (because a name hadn't been chosen at the time of registering the birth). This leads me to the conclusion that these two children, born in England, were not registered (yes, it did happen

and means there is no birth certificate available).
Shaun
Incidentally, I see that Patrick and George, who were with their father on the 1861 census, are together on the 1881 census with Patrick shown as a visitor in the family home of George with his wife & children in Northfleet, Kent.