I would give up 2 months after closing date. (not application date). My guess is they typically shortlist after it closes and then rank the candidates after interview. So there is a chance that your 2nd round may come 2m later if their first choice rejected, but unlikely that they shortlist first round applicants so late. So far I received interview requests 1-2 weeks after closing date, with one being 1m after.

As for how many rounds, it depends on the agency and interview panel. If your interview panel includes both deputy director and director the first time, it will be only one round. If it is just the AD or DD, then there will be 2nd round. If you applied for DD role, then one level above D is needed.