In general, you can partition the loop to delegate to "workers" and have each instance pipe the output to different files, each corresponding to an output stream. This avoids the need for mutual exclusion around your output streams. If you need to aggregate logs then run some log aggregator.

Maybe not nohup itself, but I often do simple background tasks + a blocking `wait` statement in a loop, for instance here: https://github.com/chollinger93/debian-scripts/blob/master/u...

If you don't use nohup, you can just pipe stdout and err to a different file descriptor for logs within said loop.

Or, of course, script something in a language of your choice.