Since in your question you said it's a PHP script, maybe the best solution could be to simply add in your script:
ignore_user_abort(TRUE);
In this way even if wget
terminates, the PHP script goes on being processed at least until it does not exceeds max_execution_time
limit (ini directive: 30 seconds by default).
As per wget
anyay you should not change its timeout, according to the UNIX manual the default wget timeout is 900 seconds (15 minutes), whis is much larger that the 5-6 minutes you need.