There are varying levels and types of qualifications you will need for different types of jobs in government statistics. For some jobs you will need at least a 2:1 in a degree with at least 25% statistical content (e.g. statistics, psychology, maths, economics). For other posts, for example statistician posts in the GSS, you will need to hold a first or second class degree in a discipline containing formal training in statistics.
Typically, you will enter at a level which is in line with the overall civil service grading; for example ONS recruits at the RO (Research Officer) and SRO (Senior Research Officer) level, with RO being aimed at recent graduates and SRO at those with more experience or higher qualifications.