The very short answer is that if the ticket is flexible, you should be able to change it online (hit and miss), or by phone or at the airport. It really depends how easily you have access to each.

I had a ticket changed to the flight I wanted to be on at T5 at the check-in desk at 46 mins before departure, when the cut off for check-in was 45 mins. The check in agents CAN do anything that they want to a booking (including in that case putting me on a flight for which there were no seats in the lowish fare bucket that I had paid for). Overall the check-in desk is you best option, but recognising that if flights are busy and you don't call up in advance you may miss out on seats that disappear between when you could have called up and when you get to the desk. It really is no more than common sense !