As mentioned in our rundown of the news from Activision Blizzard's Q2 earnings call, Overwatch will be heading into a beta "this fall." Based on past experience, that's likely to be closer to something of a soft launch, as it was for Hearthstone and Heroes of the Storm. Fall could technically mean any time between now and December, but September-November is the more accepted meaning. With the current release schedule for Blizzard-things, I'd wager late in that first month and possibly even sooner.
To be clear - they could launch a beta that was both fun to play and useful to them tomorrow. Based on videos and experience at events it's ready to be played, though may still be buggy and imbalanced. The timing is about polish and opportunity - when is it not going to get in the way of other Blizzard games,and when will it not have the news of its launch dominated by other releases? By that logic you can probably eliminate, for example, September 1st, being MGS5's launch date.
It's possible we'll hear a date at Gamescom, if they have one, but that would be uncommon for Blizzard. Usually these betas are not announced to be starting until very shortly before they do, as the decision on exactly when to actually push it live is made semi-last-minute. What we're likely to see is the announcement and showing off of at least one more new character, including the roller-blading brazilian musician Lúcio teased last week.
The earnings call also mentioned that Overwatch is seen as a very likely area of e-sport growth for the company. Deliberate or no, Blizzard have a talent for making games that don't target e-sports and then become massive ones. Hearthstone, WoW and Brood War if you want to go back that far. Overwatch clearly isn't targetting it as heavily as something like SC2 or Heroes of the Storm did, but if they're concious of the fact it could be one, that's the best of both worlds.
Naturally, we'll have all the news from Gamescom about Overwatch, where it's playable for the first time in Europe. The media briefing is at 11am UK time tomorrow, though it's unclear if it will be streamed. The main Blizzard stream starts the next day and OW is noted to feature heavily on Saturday at 5pm.