Concerning to GIG, nothing will change as Lan Ecuador does not run any Brazilian Flight. Lan Argentina only fly to Sao Paulo now, keep plans for EZE-GIG non stop in the near future. If BA would like to use Lan network, it could do it now as Lan nowadays runs 5 flights to Brazil (3 SCL-GRU-GIG, 1 SCL-GIG and 1 SCL-GRU). From Brazil there's a light difference: 2 GIG-GRU-SCL, 2 GIG-SCL and 1 GRU-SCL.
I would say BA will keep it's flight unchanged. But for sure they could fly LHR-GRU and LHR-GIG-EZE.
New York + Rio de Janeiro = One of the best combinations !