Video stream with delay
What you are looking for is called a Broadcast Delay Processor. Unfortunately, these processors come as a hardware device that you feed your signal through, and therefore they are relatively expensive.

I did look around for a software based broadcast delay for you, but I was not able to find any.

Luckily though, you can use VLC (free and available on Linux and Windows) to stream with a delay. Check out this command:


  Display stream output

  --sout-display-delay=&lt;integer [-2147483648 .. 2147483647]&gt;
                             Delay (ms)
      Introduces a delay in the display of the stream.



In your case use it like with this value:  --sout-display-delay=30000

You can also record your webcam with VLC, see this article for details:
Recording webcam videos with VLC Media Player

In other words, using VLC with the provided commands you should be able to both record your webcam and stream it with a delay (at no cost).