I'm afraid it could come under either Lewisham or Greenwich, as Blackheath straddles both.
However, on freebmd, using his full name, and those two districts, there are only two between 1858 and 1862:
Sep 1859 HARRIS George William Greenwich 1d 517
Mar 1860 HARRIS George William Lewisham 1d 724
You can either purchase both, or take a gamble on one. Unfortunately both would result in the same age calculation on the census. There's really no other way, and even this has no guarantees, as the index may not be in his full name at all. Sometimes you just have to take a chance.
You didn't say which census you found him on. It might help if we could look at that. Do you have the reference details? And about his fathers name - is that a definite for Walter?