3 replies (most recent on top)
You should definitely apply right away, but there may be a delay while the paperwork works it way through your state's system; the first UI check may be delayed but will cover back to when they determined you became eligible.
Each state has its quirks that will determine when you actually become eligible to collect. For example, in some states severance is treated as salary continuation and delays when you become eligible to start collecting UI; in some, it depends whether or not receiving the severance is contingent upon signing a waiver of claims.
You can and should apply immediately. You might not start getting money for a week or two, but you should apply right away.