I've tried to do this in the past with some success (radio call in front of TV action). There is a piece of software that will delay line in audio for a computer. My method was to use 2 computers (or at least 2 sound cards) to do this. But this was before the days of tablets/phones that could send the signal (I swear I'm not old enough to be using that line). Nowadays you could run the audio off of a device and into the line in from a PC/laptop through the sound delay software and out to speakers. I don't recall the name of the software, but I still have it installed at home. I can find the name if interested.
The problems with doing this were that A) it was at least a small pain to sync up because the delay would have to build up in the program (typically 6-9 seconds) and any change would reset, so it would take a couple minutes to complete and B) the audio sync would drift (either from tv or radio) and so the delay would become large (seconds) over the course of the game and would need to be redone at least once.
Also, it may be possible that something similar exists as a tablet/phone app or that a better program has come along since then (2005ish). But software solutions do exist. Another option is a hardware delay.
Edit: here is the software I used - http://www.daansystems.com/radiodelay/