Instead of using cron (either manually or waiting for it) you could use:
Print the current simplenews mail spool count
Send the default amount of mails, as defined by the simplenews_throttle variable.
drush simplenews-spool-send 100
Send 100 mails.
drush simplenews-spool-send 0
Send all mails.
Alternative cronjob usage
If you can't use drush you can trigger cron manually at
/admin/reports/status/run-cron, but this will also run all other cronjobs. As an alternative you could install and use elysia cron or ultimate cron (currently the later looks like to be the more active version). Both modules will allow you to run specific cronjobs manually and separated from others.