No baggage for as your basic economy fare is starting with a BA check-in. Status doesn’t provide you with anything.
On the way back, as you’re starting with American you’ll get your OWS baggage allowance. In America they don’t follow IATA MSC rules.
On through-checked baggage, USDOT regulations require the marketing carrier of the initial segment baggage rules apply to USA originating (or ending, iirc) flights. E.g. flying QF as an AA codeshare, or QF connecting from AA marketed by AA, AA rules would apply. These take precedent of IATA rules if travel begins (or ends) in the USA.
FAQ: Baggage / Luggage Rules, Allowance, Through-check on AA & Other Airlines
If you connect on the way home (e.g. YUL - ORD / ORD - LHR on AA and then AA or BA respectively) the you’ll get your OWS allowance.
I frequently fly USA to UK on basic economy tickets and when starting on an AA flight in USA, I’ve always received my OWE allowance, and the bags have always been checked through to the UK. In order to receive said allowance, I frequently connect AA to BA internally in the USA.