(Throwaway account, YMMV) We applied on 4/8 and received our PPP on 4/15.
We have almost 40 employees and bank with Chase and Wells but we realized early on that the big money-center banks were a couple of steps behind are were going to be overwhelmed.
Eventually after applying with a few we got furthest with by applying via Kabbage and SmartBizLoans. Our company had an existing relationship w/these lenders and they seem to be the most tech-conscious of the bunch.
Our application via SmartBizLoans was approved on 4/13 and the funds deposited on 4/15. SmartBizLoans acts as a broker to a number of smaller banks which might explain why they were quicker. AMA, glad to help.