Our legal review takes 5 days, max. Who knows how long it will take to have an actual contract, but it won't be that long. They may "negotiate" with all the bidders and ask for a best and final offer. Once they accept a proposal, they have to notify the unsuccessful bidders and probably debrief them, yada yada. After all that and once legal approves, its just a matter of the CO and Contractor signing the Contract. It'll happen pretty fast.
Next would come "Notice to Proceed" and that is a bit different for Design Build. They may have a NTP for design and one or more for construction. No breaking ground without a NTP. No NTP without bonds, insurance, permits, etc.