Hi,
Yes, Walter Willoughby, the son of John Willoughby and Mary Gillatt (sometimes spelled Gilliott) later joined the Royal Navy as Walter Willoughby Gilliot (assuming the surname of his mother upon enlistment but always using his birth surname as a middle name after that.
Walter was an older brother of my grandfather Sidney G. Willoughby, so therefore a great uncle of mine.
In 1901 Census he appears with the Royal Navy as Walter W. Gilliot, a stoker on board HMS Warspite, born Masbrough Yorks, age 22, single.
In 1911 Census, he had already completed his 12 years service with Royal Navy and was living with his mother and youngest brother in Rotherham and working as a pumpman at the Rotherham Main colliery. He is listed as Walter Willoughby, age 33, married to Alice.
Subsequent to that he joined the fledgling Royal Australian Navy in 1913 as a leading stoker and eventually stoker petty officer, serving with the Aussies during World War I. His Royal Navy and Royal Australian Navy records (which I have attached to my family tree) all reflect him as Walter Willoughby Gilliot. City directories in Sydney Australia where he and Alice settled after World War I, and adopted a son who was named Ronald Arthur Gilliot, all reflect that name as does his Australian death record in 1960 at age 81.
Additionally, my grandmother (wife of Sidney G. Willoughby) told me that Walter had changed his name and she was never wrong. She also said he served in the "Navy". It is also true that my grandfather (with the Royal Flying Corps / RAF), another brother with the Royal Field Artillery and another with the Kings Own Yorkshire Light Infantry all served, as did Walter, during World War I. So 4 of the 6 brothers served in WWI. The 2 eldest were "too old" according to my grandmother, to serve.
Incidentally, my father's middle name (Walter of course) and therefore mine since I am a junior were given in honor of Walter Willoughby Gilliot. This is another fact handed down by my grandmother.
I have seen the discrepancy in birth dates and noticed that the December 15, 1878 date it is identical to a brother in law in the family - one who was married to one of the sisters. This is probably nothing more than a coincidence though. I will take a look at the baptism records - perhaps that is the only record that Walter had at time of his enlistment.
Sydney Walter Willoughby Jr.