According to the 1911 census image, Joseph was born in England. Do you know where he was born?
This will help narrow down his military record, if it still exists. Find My Past have many pre-WW1 military service records.
Also, his occupation on daughter Mary Elizabeth's 29 July 1906 birth record is "labourer". When he married on 13 June 1906, he was a soldier. So it suggests he left the (regular) army before his daughter was born about 6 weeks after the marriage.
UPDATE: I've found a record for a Joseph Mead, born 1881 in Bushey, Berkshire. Record indicates born near the town of Rickmansworth. Father's name: Daniel Mead. JM served initially with the Royal Fusiliers, then the 13th & 19th Hussars.
His military record indicates he attested in 1899 and was transferred to Army Reserve on 25 July 1906. Could be a coincidence, but that date is a few days before the birth of Mary Elizabeth. His 5-page record is on Fold3 which I have access to.