The first battle I hadn't checked my XP going in. (I had expected an XP text line at the end.) We won, but I had no idea how much it awarded.
Second one, lost. Got 970 XP though.
Third one, won. 1,733 XP (woot!).
At most, the battles take 20 minutes. If one side gets three flags first, then it ends. So, depending on win/lose status and color of the quest you do, this isn't horrible XP. Tonight I did two yellow quests that took me an hour to finish and got 2K reward from the quest and a good deal more for killing the mobs. Three WGs tonight and I got almost 5K (assuming I got full reward for the first one). However, over lunchtime I did something like 8 green quests and got about 7K XP from turning them in.
So really, not bad at all. I find doing quests stressy as hell (will I ever, ever, ever lose my FFXI fear of dying? IT DOES NOT MATTER IN WOW! Gah!), and I enjoy BGs, so this is a nice option. I'm not going to do all one or the other, but it's nice to have a non-quest option.
(Also, I have no idea if something more than just winning or losing comes into play on XP reward. That'd be an interesting thing to know, but my quick googling came up with nothing.)
Edit: Hm. I totally forgot about the money/drops/rewards you get for doing quests, and my little rogue is already way, way hurting for money (he's 18 and I only just upgraded his last bag this morning, now he's broke again). No herbing/skinning in battlegrounds, and I bet there's no skilling up of weapons. Hmmmmm.