HM Treasury does not hold data in the format required to answer the question on vacant positions for a week or longer in each year since 2010. However, HM Treasury publishes recruitment data in the Annual Report and Accounts, which is available at: https://www.gov.uk/government/collections/hmt-annual-report
Recruitment data for 2015/16 can be found on page 78.
Turnover data for HM Treasury can also be found in the Annual Report and Accounts. The information for 2015/16 can be found on page 79.