The total time from the signing of terms of business until the selected candidate actually starts work depends mainly on the successful candidates notice period. At certain levels of seniority 3 months would not be unusual and as such the elapsed time would be around 4.5 months. Should the resignation period be only 1 month, then the total elapsed time should be 2.5 months to start date.
We would undertake to get a shortlist of qualified candidates within 4 weeks after which there is an interview process, which could take a further week (provided you are available). By the time second interviews are complete, offers are drafted, considered and accepted by the successful candidate, a further 3-4 weeks has normally elapsed.