No points unless you actually fly. But look on the bright side...you're probably not giving up a ton of points because the deep discounted BA tickets will earn only 25% mileage credit, and even worse - 5% EQD's. The fare buckets are listed below with mileage earnings. You'd have to check your e-receipt to see the actual fare bucket you're booked in.
https://www.aa.com/i18n/travel-info/...sh-airways.jsp
But have you asked BA if there is any kind of residual credit towards different dates/same city pairs?
I had a similar situation on the cheapest available BA fare. No refund, no changes to flight segments. But the actual dates could be changed for a fee and fare difference.
OP mentioned Business Class. That's 25%EQD, 1.5EQM and 1.25RDM.