I personally suspect the changeover started August 1, and end of October would be 90 days (give or take) later which seems to be, from the various contract experts on tumblr, the agreed-upon standard for contractual changeover periods.
The reason I also posit the beginning of 2015 is because Azoff Music Management currently has a non-compete in place which disallows them from taking on new high profile clients until it expires new years day 2015 (barring certain circumstances that we have no access to information about).
As for OTRA, the old team would get a cut of anything they organized, but the new team would be the ones actively supervising the tour. IE, some Azoff or Azoff employee would be on tour with them, not Modest representatives.
Be clear, One Direction will have ties with Modest as long as they exist as a band (if not longer). Modest owns a large portion of the publishing rights on their songs, especially from the first two records, not to mention the boys joint investment in 5SOS with Griffiths, McGee and Bloomfield. If they leave, they have to leave on decent terms lest they fuck themselves out of future earnings.
They cannot burn bridges and Modest wouldn’t want to anyway because they don’t want Irving Azoff as an enemy in the event they want to break any other acts they have into the US market - seriously, go read about him, the man in incredibly powerful. Gain his ire and YOU ARE FUCKED.