British Military in Canada 1850's

Hi Everyone

I'm researching my ggmother Elizabeth Creamer she was born in NewFoundland in 1863( cant find any records to say this is true yet) But i have just found out recently that her father was in the 39th regiment from Ireland and that the regiment was in Canada around that time. If that is the case would her birth have been registered in the Canadian births and deaths ???
Elizabeths parents were a James Creamer and and his wife was Maria they were married in Dublin Ireland in 1852 and they had 5 children Elizabeth 1863 and Mary born 1865 both girls born in Newfoundland, there was also 3 boys ( not sure where they were born) apparently they died on the voyage out to Australia around 1874.. Maria the mother somehow ended up back in England because she married again 1870....Cant find this family in any shipping lists either going to Canada or leaving or on any census in Canada..
Did the british military have to participate in the census at all ?
Did the british military have to register the births in Canada ?

Cheers
Robin