Some correction on the screamer radio directory list: different address (e.g. for Sports Extra) listed on the site, but still - they work in VLC.
Interesting question, but the immediate question popping up for me is: wouldn't that mean the player actually has to save the stream in some form?
Screamer for example also has a record button, and well, of course, after recording you can go back and forth as you like. But usually the stream isn't saved, right? It's just played.
Just took a BBC 1 TV stream (you can find that in the UKTV thread in the Live TV section here), opened it with VLC, paused, and that strangely works. No jumps or the like, no recording necessary.
If that works on TV, it SHOULD work on radio as well. Could be a question of what the stream is made of or how it is streamed (or the protocol).
A quick note on VLC's record button: this is not shown standardly, you must check the "advanced controls" under "view" to get it.