It is still technically possible to sort this - the Belfast passport office has same day appointments available for Monday morning. A new ESTA can be applied for as soon as s/he has a new passport and usually only takes a moment. It could - in theory - all be fixed by Monday evening in time to travel...

The OP's friend needs to decide whether the cost/hassle of going to Belfast for this on Monday is worth it to avoid the cost/hassle of forfeiting the BA flight.

I agree this is in no way BA's fault.

(And I can't believe this is the post that made me terminate my long term lurking!)