add_action( "bp_core_activated_user", "bp_autologin_on_activation", 40, 3 );

	function bp_autologin_on_activation( $user_id, $key, $user) {

		global $bp, $wpdb;

		//simulate Bp activation

		/* Check for an uploaded avatar and move that to the correct user folder, just do what bp does */
		if ( is_multisite() )
			$hashed_key = wp_hash( $key );
		else
			$hashed_key = wp_hash( $user_id );

		/* Check if the avatar folder exists. If it does, rename it, move it and delete the signup avatar dir */
		if ( file_exists( BP_AVATAR_UPLOAD_PATH . '/avatars/signups/' . $hashed_key ) )
			@rename( BP_AVATAR_UPLOAD_PATH . '/avatars/signups/' . $hashed_key, BP_AVATAR_UPLOAD_PATH . '/avatars/' . $user_id );

		bp_core_add_message( __( 'Your account is now active!', 'buddypress' ) );

		$bp->activation_complete = true;

		//now login and redirect
		wp_set_auth_cookie( $user_id, true, false );
		bp_core_redirect( apply_filters ( "bpdev_autoactivate_redirect_url", bp_core_get_user_domain( $user_id ), $user_id ) );

	}

Gist link: BuddyPress autologin on activation