For information on the 12th regiment, see http://www.austbuttonhistory.com/?p=8080&preview=true
14th (2nd Battalion Buckingham) Regiment of Foot
The 14th regiment of Foot arrived in Hobart, months later than expected, on the 31st October 1866. They Served in Adelaide, Hobart, Melbourne before leaving in 1870.
The button can be seen on Cossum page 64, also at https://collection.maas.museum/object/85036
The 77th (East Middlesex Regiment) of Foot.
Two ships bearing the 77th arrived in Sydney Harbour 0n 27th September 1857. However, they were not here long, leaving Sydney for Calcutta in April 1858. It was felt there was need for troops in India, and that “the great majority of the troops in NSW are not required here”.
In there short stay in Sydney, there was some reports of drunkiness and theft, and problems with the local police:
See Cossum page 67. Also https://www.worthpoint.com/worthopedia/77th-foot-east-middlesex-regiment-1777688768