start date advice

hi!! I was recently offered a place in a department within the civil service, in the letter they said that the department would be in contact regarding start dates etc, but it required extra security clearance. this was sent about a week ago now and I have not heard anything from the department regarding start dates or extra security clearance. I passed vetting clearance approx 2 weeks ago now as well. just was wondering if anyone has experienced anything similar and if anyone knows how long it might be until they contact me? i'd need to give a months notice at my job so I'm thinking I should defo wait to hear from the department before handing in my notice.