Small depositors (under US$250K) can get paid off quickly if the FDIC takes over, but larger ones may have to wait until SVB's loans are sold off to other banks for cash.
The collapse of a solvent but illiquid bank is well worked out in the US. Here's a 60 Minutes episode where they got to cover the process.[1] It's not too bad for the customers, but it is really bad for bank management. The 60 minutes video shows the moment when the FDIC people show up and tell the CEO he's finished.