Really impossible to say for certain without any insignia, however with the name Mackenzie then most likely a Scottish Infantry regiment. Also the fact it was found in Glasgow would make it most likely to be a lowland regiment, so the order of local regiments would be the Highland Light Infantry which is the local Glasgow regiment, or possibly the Cameronians or the Royal Scots Fusiliers who both also recruited from Glasgow and surrounding areas. I would say Cameronians but then I'm biased. The tunic below would have been worn in South East Asia but as in my uncles case the Cameronians also fought in North West Europe.
This battledress blouse was worn by Sergeant Frank Picken from 1944-1947. The cloth badges on the sleeve indicate that Sergeant Picken was a Cameronian who served as part of the South East Asia Command (SEAC). The SEAC’s operational area during the Second World War was in India, Burma, Ceylon, Malaya, Singapore and Sumatra. Service records show that Sergeant Picken served in India and Singapore. The picture badge is the formation badge of the Allied Land Forces South East Asia Command which was previously known as the 11th Army Group of the SEAC. The three stripes or chevrons on the sleeve indicate the rank of Sergeant. This tunic is on display in the Cameronian museum 129 Muir Street Hamilton, ML3 6BJ. The museum holds a collection of objects relating to Sergeant Picken’s service with The Cameronians in the 1940s. Picture credit the Cameronian museum, Hamilton.