You may be able to start in the Civil Service, for example, as a higher executive officer. You could then apply for a place on the Civil Service Fast Stream scheme, which can eventually lead into a management career.
Each department sets its own entry requirements, which often includes a university qualification. As an existing employee, you may be able to apply without a degree, though this varies between departments.
After completing the scheme and gaining experience, you could move into a management role as vacancies become available.