my experience with Tower was, with in seven days you should be OK provided you already approved and submitting POA, MPV , Insurance Copy (all these need VIN). Tower send you check overnight provided they close the loan process before 3pm on that day. on the closing day, tower call you and go over the terms and conditions (Apr rate, auto debit etc... and if this happens before 3pm and next day morning you can expect check through fedex).
Generally Tesla will be calling you to confirm the delivery date and my case was, initially tesla gave me chance to pick the delivery date and time and later on i asked SA to push the date to another 4 days. I also got tesla automated voice and option 2 is to change the delivery date (option 1 is to confirm the date and time, 2 to change and 3 to speak advisor ( most cases you never get a call from advisor)