corporate-wage-slave

Yes the critical aspect to this is that rebooking guidelines won't normally allow you to add stopovers, stops over 24 hours. They do generally allow layovers, transit stops under 24 hours, so long as you aren't too many extra miles added as a result. Adding NYC and/or ORD would be OK there. So adding new stopovers into your trip would normally result in a re-fare. You may be able to argue for a stopover as an allowed "going outside policy" if for example there are only 3 flights a week or some such restriction, or flights are totally full, which won't apply here. However, many BA fares do allow unlimited stopovers, or charge low fees for a stopover, so you may be able to get an agent to rebook you to the new trip via NYC, with layovers, and then ask for a re-fare quote for adding the extra time in NYC. It could be fairly modest, and zero if you ask this after flying the outbound sector on two-way journeys. I would not expect the last option, AMS, to be allowed since it involves going through LHR rather than as an alternative to LHR.