Sterling Times said:

It is possible to call up the on-demand programme
and dial-up any elasped time without pause.

I noticed you can do this with the "listen again" facility of the (Liverpool) City Talk website. It appears to be a stream of a file rather than a download but you can scroll along to any point in the duration of the programme and start the stream from there.

I don't know how it is done though as all of the streaming experiments I have done have been encoding and streaming what is effectively a live source (even if it is in the form of a recording that is playing in some form of media player). If I do stumble across anything about it I will post it here.