I’m a bit confused about upgrade with Avios pricing and technique.

I’ve booked a 125 plated itinerary SOF-LHR-JFK/JFK-LHR-AGP with transatlantic legs in T and European legs in K classes, based on a r/t T fare SOF-JFK with the open jaw at the end.

Is BA availability in business class for 57.5k miles on the AA website a proxy for BA upgrade availability on that same flight using Avios?How can I see what the upgrade pricing in Avios would be? ExpertFlyer’s search feature breaks and seems only to let you search for V class availability (extra redemption seats for Gold, or something like that) but doesn’t let you specify U or another letter class even though the search page originally shows some boxes for you to search for any letter booking class you want to specify.

Do BA process mileage upgrade between origin and destination, (so for the first set of flights SOF-LHR-JFK) or is it segment by segment?

Must I call or can I do any of this self-service online?

Sorry for the silly questions — never done this before. I know there’s probably an FAQ somewhere.