score:3
As explained in another thread on FlyerTalk, your baggage has cleared customs in Shannon so it may be transferred onto onward destinations. No further examination by US Customs is required at JFK.
However, the layout at JFK Terminal 7 means that there is no easy facility to take bags onto the inter-terminal transfer system from the gate that BA 1 uses. Nor was onward baggage handling considered a priority when the BA1-4 service was launched (it is not intended for connectors). So instead all the bags end up on the domestic reclaim carousel.
After about thirty minutes any remaining bags on the carousel are collected by ground staff, and at that point, forwarded over to the appropriate terminal for onward flights. Bags are moved between T7 and T8 approximately once per hour.
The minimum connect time between T7 and T8 is two hours (even if arriving precleared), so in principle your bag should make your subsequent AA or US flight. However you can see that the process has some possibility of failure.
Therefore you may wish to grab your bag off the carousel if you see it and transfer it on the air train yourself to T8. You can simply hand it over at the First check in desk with a quick explanation.
The alternative method mentioned in the thread (I've never tried this) is to talk to BA Special Services as soon as you get off the plane who will ensure your bag is interlined immediately instead of sent onto the domestic carousel.