For now it's limited to domestic flights, so BA won't be affected, though anyone with a connecting intinerary from BA to a domestic AA/AS flight could be.

But of course if the shutdown continues it will likely mean more reductions and at some point international flights will be affected. If and when BA do cancel flights they'll be obliged to refund/rebook per UK261, but afaik there's no pre-emptive policy like UA has.