Responses report with all fields times out - error message:

Gateway Timeout
The gateway did not receive a timely response from the upstream server or application.

Report without details and Simple Mode does not cause problems.

Provider: SiteGround
LimeSurvey: 2.50+
PHP 5.6.21

Any hints would be much appreciated.

Unfortunately phpinfo is too large to attach

Did you get a solution for this? I am getting similar timeouts, especially when clicking submit after completeing the survey form.

Also hosted with SiteGround

My first guess would be that either the limit for execution time or for memory in PHP is too low on your server. At least for abdagon's problem, as creating statistics for all answers can require quite some resources on the server.
Now when submitting a survey, I am not sure if this is the same problem.

Thing is, the survey is logged and submitted OK. The customer even gets the confirmation email saying he successfully submitted the survey and that it was saved. It's just the on screen message after hitting submit is not displayed; the timer just spins for ages and then you get the Gateway Timeout