I went through this on the phone recently, the process was:
1). The agent held my new flights for me.
2). She cancelled my previous flights.
3). She called someone internally to expedite the mileage refund and check it would all be ok.
4). She confirmed my new flights to me.

For some reason it then seemed to take about 24hrs for my new flights to confirm correctly in MMB which I think was due to the miles being re-deposited.