How much authority did Westminster have over the government of Hong Kong?
Also, I am aware that non-British Hong Kong residents were not given the right to elect their governor, but were they given any other political power?
Similar to many countries, there's three branches of government in Hong Kong: Executive, Legislative & Judicial. The key figures in the Executive & Judicial branches have always been appointed by the State during the Westminster times.
However, the Legislative Branch, Legislative Council of Hong Kong (LegCo for short) did hold its first election in 1985. The 1985 LegCo consist of 56 members with 10 officials, 22 appointed members, 12 elected by [functional constituencies](http://en.wikipedia.org/wiki/Functional_constituency_(Hong_Kong)), and 12 elected by electoral college.
After many election reforms, Hong Kong held its first and last fully elected LegCo under the British in 1995. The 1995 LegCo consist of 60 members with 30 elected by functional constituencies, 20 elected by direct elections, and 10 elected by election committee. Though, the appointed Governor was the President of LegCo until 1993.
source straight from the Legislative Council official site with a nice table on the last page.
ps. Generally, I just lurk /r/AskHistorians. However, there's no answers for 4hrs, so if I did anything wrong please let me know. Thanks.
Follow up: To what extent could it be said that the British Government withheld elections so as not to annoy the Chinese? (might be a leading question, I guess I'm asking about motives for any withholding of Democracy)