Email campaign and 'mail to group' messages are not sent instantaneously.
If email is to be sent to less than 50 contacts, emails are queued in the Outbox and sent out with up to 5-minute delay.
If email is to be sent to 51+ contacts, emails are sent out with up to 15-minute delay.
If you need to send an instant automatic message to contact, use Autoresponder functionality instead of (or in addition to) email campaign templates.