Store your configurations on EFS, and mount EFS in the location Nginx configurations are expected. Alternately put them on Amazon S3 and run a sync occasionally, or use s3fs (beware s3fs may not be good enough for production use).
When you need change your configuration, increase your autoscaling group desired size to double what you need to trigger new instances with the new configuration, and then back down to what you need which will remove the old instances. Alternately just do a rolling reboot of the servers.
Another option is to just push the new configurations to your server using a basic automation tool, like AWS code deploy.
The fully automated options above are technically better and cleaner, but if you rarely change configurations and want an easy solution this could help.