If the flight is within 24 hours and you don't think you are well enough to travel then you contact BA. Otherwise it's Amex. Generally for this sort of thing it's best to rely on insurance (or self insurance if that's more cost effective) but BA does have some policies in this space that can be helpful, depending to some extent on what your next steps will be. Amex - and all travel agents - can act as a go-between here, though in some cases TAs will cook up their own rules and processes around this. Note that depending on status there may be paperwork involved (e.g. doctor's note, fitness to travel note).