"By April 15" carries with it the implication that it may be *close* to April 15. So I wouldn't say "could happen any time, starting now", I'd say, "will arrive on or before April 15, but probably close to that date."
"will arrive on or before"
Pour dire "vers (date/heure)", je crois qu'on dit "around (date/time)". Pour les heures, on dit familièrement "at twoish" (vers deux heures). Du moins au Royaume Uni.