Yes there are loads of opportunities aside from the fast stream. Here's the careers page for my role
https://gss.civilservice.gov.uk/people-and-careers-2/statistical-officer/ I'm a stats officer working in the Office For National Statistics (ONS) so I'm afraid I can only really tell you about a very small part of the civil service. I applied during my final year of uni (I did psychology) and started 2 months ago straight after graduating.
I work in Administrative data, which is basically data that wasn't collected for the purpose of statistics, so I do a lot of liaising with other departments such as HMRC and DWP to see if we can share their data to save money/time and get better quality stats for the government to base their decisions on.
I'm not sure how many graduate, but not grad scheme, jobs come up outside of the stats departments but definitely keep an eye out here
https://www.civilservicejobs.service.gov.uk/csr/index.cgi Let me know if you have any further questions