Yes, you legally own the property once it's "completed" - Full purchase balance paid. Your solicitor should contact your mortgage provider and handle it all for you. If you're happy with everything, then "exchange" as soon as possible as this will seal the deal so to speak. This won't guarantee you'll make it to completion but it makes it far, far more likely and also gives you legal recourse if it fails (depending who's at fault for breaking the contract).