Summary: Doesn't matter if spawn timer says 5 or 30 seconds. Warpgate is exception (works immediately). Reproduction steps: 1) Be killed (by enemy, ally or by yourself). 2) Try to spawn a vehicle from the map.
Repeatability: Always. P.S. Also, quick vehicle spawn is affected by 30s acquisition timer (a cooldown between spawning two of the same vehicle), but it isn't mentioned anywhere on the map.
Adding a detail onto this: If you redeploy manually (rather than being killed by enemy fire), you can spawn a vehicle from any eligible allied base on the map within 3 seconds.