If you use the online tool to book the ticket, it will be issued by whoever is the marketing carrier of the first segment (with a couple of exceptions - for example, QF does the ticketing for JL, AA does it for RJ). If your first leg is LHR-JFK, it will be ticketed either by AA or BA, depending on whose flight you choose (and note it is the marketing carrier, not the operating carrier so (in theory) if you take a BA codeshare on an AA-metal flight, it will be ticketed by BA). Conventional wisdom is to avoid having BA issue the ticket if at all possible, as they tend to charge punitive fuel fines which others (e.g., AA) do not.

As to problems/delays with points posting, there shouldn't be a problem ..... though having said that, the AA miles never, ever post automatically for the QF legs on my AA-issued xONEx tickets. AA # shows up on the QF boarding pass, no issues with status recognition etc., but the miles never post automatically. No big deal really - just have to keep the boarding passes, follow up etc.