I think I'd start the search with the marriage of Patrick and Annie to confirm details of fathers and then work from there. There are over 250 births for a Patrick Casey in Ireland between 1875 and 1885 (to allow for inaccuracies in reported age), including at least 15 in registration districts which covered parts of Co. Waterford, so additional clues could be important in narrowing down the correct birth for 'your' Patrick.
The 1911 Census of England shows a good match for your Casey family at Oakfield, East Ham, London with parents Patrick Joseph Casey born Waterford about 1880 working as a gardener (domestic) and his wife Annie Eveline, two sons, including Walter, and a daughter. The three children and Annie are all shown as born in England - so a marriage in England seems likely. The census form indicates they are married about 6 years and there's a good match around the right timeframe on the BMD index in 1904, Lewisham Registration district. One of the cross matching names is an Annie Eveline Weller. The index references are :
Name : Patrick Joseph Casey
Year/Quarter : 1904 / 2
Registration District : Lewisham
Volume : 1D
Page : 1929
A check of the Births index entry for Walter's confirms that his mother's maiden surname was Weller.
You can use these references to order a marriage cert from GRO England/Wales A marriage cert. will include date and place of marriage, details of the bride & groom and where they were living, along with the names and occupations of their fathers. Also included on marriage certs are the names of witnesses, which can relations such as brothers, sister etc..
p.s. have you located Patrick Casey on the England 1901 census ?
The 1939 registers shows a possible match for this Casey couple, still living in Lewisham, Patrick and Annie E. Casey presumably his wife - both are shown as married but relationships are not shown. Patrick's occupation is shown as 'Stoker Heavy Worker' so if it's the same couple he changed jobs. A date of birth is included - for Patrick it's recorded as 29th June 1879.
I checked for possible births in the Co. Waterford area matching that date and found no exact matches, however there was one very close to that date - a Patrick Casey, born 25th June 1879 in Waterford city (2nd entry on the register image) Place of birth and family residence looks like Barronstrand Street, parents are William Casey and Eliza White. Father's occupation is fairly distinctive - a Tin Smith - hopefully that detail along with Patrick's marriage cert will confirm the match for you. This couple also had at least one other child, Alice born 27th October 1880.