The Contract and Transfer on a sale is usually signed earlier than on the purchase, but it will sit on the file until your buyers have signed theirs, which they will get when their solicitor "reports" to them.
Your solicitor will also need to "report to you" on your purchase. This is when you will get the Contract to sign for your purchase. The report stage is generally when they are in receipt of all paperwork, all enquiries have been dealt with satisfactorily (or at least most of them) and they are nearly ready to exchange subject to everyone else being ready too.
The report details all the information the solicitor has found out about the property. You should get copies of the enquiries they raised and the replies, together with other relevant copies of documents about the property (i.e. searches, cerificates/warranties, mortgage offer and deed to sign). The report should also include the Contract (and mortgage deed) for your signature, along with the SDLT form (unless you are instructing them to fill it in on your behalf).
Once the solicitor has all the signed documents back from you along with anything else they've asked for, they will need to review the file to make sure everything they need to exchange is there, as well as check that your buyer's solicitor is ready and your seller's solicitor is ready. They will not be able to exchange without the signed documents.
They will then ask for a preferred completion date which has to be agreed between all parties and then will request your authority to exchange. The buyers at the bottom will need to transfer their deposit to the solicitor.