Work won't start on your purchase until solicitors are instructed on both sides and the purchase solicitor receives a draft contract from the selling solicitor, together with various important papers, i.e. title plan, deeds (office copies), fixtures & fittings list, seller's property information form, EPC and leasehold information form (if leasehold). It very much depends on how quickly the vendor returns the forms to their solicitor as a draft contract won't usually be forwarded to your solicitor until then.
What Spickle said. Ours took 3 weeks, but our solicitor was chasing them hard by the end and I got the impression that was a bit slow. For our sale, I believe the other side had the draft contract within a week as I'd already given the forms etc to my lawyer.