At BNA the first hit for "Bristo Technical Institute" is in June 1949 (appointment of a headaster, Peter A Baikie (46)), and, apart from a reference to the past in 1993, the final mentions are in 1967 (vocational evening classes).
1964: The College (Heriot-Watt) are leasing the old Bristol Technical College. (Unclear if this means that it has closed, or simply moved.)
1964: Disposal of two baking ovens at Castlehill School; contact the headmaster at Bristol Technical Institute
1965: no articles
1966: a single article advertising evening classes
1967: an article mentioning the success of Shetland students refers to the Bristol Technical Institute at Castlehill School of Catering.
Putting all of this together I think that the former premises were available by 1964 and some version of the institute operated from Castlehill School.