Does Brann Bronzebeard's ability activate as soon as he is played?

Yes, Brann will still work. Like every other ability in the game, Brann's ability still works while he has summoning sickness. There are two possibilities as to why Brann didn't work.

The first and most likely is that Brann only causes the battlecry to trigger twice. It doesn't allow you to select a new target for the second trigger so targeted battlecries will hit the same minion. This might be useful e.g. Abusive Sargent, or might have no additional effect e.g. Big Game Hunter. This can also apply to random target minions such as Stampeding Kodo- due to the order of effects, it can hit the same minion twice.

The second is that the minion didn't actually have Battlecry. Combo and Choose One minion abilities are similar to battlecries but for Brann's ability they are not.