In the movie First Contact, not only do they get a live feed from the battle in Earth's solar system, but getting there takes only a tiny warp jump.
Hmh? The radio feed (whether live or not, we don't know) was from the Typhon Sector battle, supposedly mere 3½ hours away.
After that, there was a cut in action that may have been two minutes long, or then two weeks. No real way of telling. Only after the cut did Earth get involved.