Does curl have a timeout?

Yes.

Timeout parameters

curl has two options: --connect-timeout and --max-time.

Quoting from the manpage:

--connect-timeout <seconds>
    Maximum  time  in  seconds  that you allow the connection to the
    server to take.  This only limits  the  connection  phase,  once
    curl has connected this option is of no more use.  Since 7.32.0,
    this option accepts decimal values, but the actual timeout  will
    decrease in accuracy as the specified timeout increases in deci‐
    mal precision. See also the -m, --max-time option.

    If this option is used several times, the last one will be used.

and:

-m, --max-time <seconds>
    Maximum  time  in  seconds that you allow the whole operation to
    take.  This is useful for preventing your batch jobs from  hang‐
    ing  for  hours due to slow networks or links going down.  Since
    7.32.0, this option accepts decimal values, but the actual time‐
    out will decrease in accuracy as the specified timeout increases
    in decimal precision.  See also the --connect-timeout option.

    If this option is used several times, the last one will be used.

Defaults

Here (on Debian) it stops trying to connect after 2 minutes, regardless of the time specified with --connect-timeout and although the default connect timeout value seems to be 5 minutes according to the DEFAULT_CONNECT_TIMEOUT macro in lib/connect.h.

A default value for --max-time doesn't seem to exist, making curl wait forever for a response if the initial connect succeeds.

What to use?

You are probably interested in the latter option, --max-time. For your case set it to 900 (15 minutes).

Specifying option --connect-timeout to something like 60 (one minute) might also be a good idea. Otherwise curl will try to connect again and again, apparently using some backoff algorithm.

There is timelimit: /usr/bin/timelimit - effectively limit the absolute execution time of a process

 Options:

 -p      If the child process is terminated by a signal, timelimit
         propagates this condition, i.e. sends the same signal to itself. 
         This allows the program executing timelimit to determine 
         whether the child process was terminated by a signal or 
         actually exited with an exit code larger than 128.
 -q      Quiet operation - timelimit does not output diagnostic 
         messages about signals sent to the child process.
 -S killsig
         Specify the number of the signal to be sent to the 
         process killtime seconds after warntime has expired.  
         Defaults to 9 (SIGKILL).
 -s warnsig
         Specify the number of the signal to be sent to the 
         process warntime seconds after it has been started.  
         Defaults to 15 (SIGTERM).
 -T killtime
         Specify the maximum execution time of the process before 
         sending killsig after warnsig has been sent.  Defaults to 120 seconds.
 -t warntime
         Specify the maximum execution time of the process in 
         seconds before sending warnsig.  Defaults to 3600 seconds.

 On systems that support the setitimer(2) system call, the 
 warntime and killtime values may be specified in fractional 
 seconds with microsecond precision.

Better than --max-time are the --speed-limit and --speed-time options. In short, --speed-limit specifies the minimum average speed which you are willing to accept, and --speed-time specifies how long the transfer speed can remain below that limit before the transfer times out and is aborted.