All of the changes for each modification appear to make it into the database, so I'm guessing the delay occurs after the database write. Additionally, the email associated with the change makes it, so the delay is probably also after the email is sent. I can't identify what change may have triggered this. My last change was to add the SourceBitBucket plugin (on 2016-05-06). There are four other subdomains on the same server, and none of those are showing timeouts.
There's nothing in the mantis log that indicates errors. I looked in core/constant_inc.php to see if there was a LOG_xxx for errors, but didn't find anything that looked relevant. /var/log/nginx/mantis.access.log looks normal, and /var/log/nginx/error.log shows the timeout errors (domain details obscured) ...
Configuration:2016/06/30 22:35:26 [error] 1150#0: *4625 upstream timed out (110: Connection timed out) while reading response header from upstream, client: xxx.xxx.xxx.xxx, server: mantis.xxx.com, request: "GET /view_all_bug_page.php HTTP/1.1", upstream: "fastcgi://unix:/var/run/php5-fpm.sock", host: "mantis.xxx.com", referrer: "http://mantis.xxx.com/view.php?id=58
Ubuntu 14.04.3 LTS
Chrome Version 51.0.2704.106 (64-bit)