Not quite. Any flight you find on AA that is a SAVER AWARD flight is bookable on BA. If not on their website and there are connections, search by segment as
@Albort suggested. If not searchabel that way, then call BA with the flight info you find for the saver flight on AA. Let's take a look at my flights to and from Santiago, Chile for the Antarctica trip I'm doing later this year. We play this game to fly international business or first lie flat seats. If I wanted to book the fights on BA using Avios or UR transferred to BA Avios and was unable to find them on BA, I'd search AA for saver awards. Our trip departs 12/25 and we return 1/6. When I search AA for a saver business award this is what I get. The only saver awards available in business/first are for 12/27-12/31. I would not be able to book a flight on AA with BA Avios for 12/25. I would not even be able to book the flight with AA miles as a saver award. I would have to book it as an Anytime Award. If I change my trip to 12/27 then I can book on BA with Avios. This is an 8 hour flight and Avios is distance based so it won't be 57.5K AVios, it is 75K Avios one way.
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My dates are fixed so I cannot change my departure. I'm stuck booking an Anytime Award. When I click on the Anytime Award I can see how many AA miles this will cost me. 150,000 AA miles per person, one way.
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When I check flights back home 1/6. I see that there are absolutely NO business/first saver awards for the entire month! Plus the Anytime Awards for my date is 195K per person.

So, I could not book this with BA Avios.
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This is why it is important to plan a couple of years in advance if you know you will have fixed dates. You need to get a general idea of how many miles/points you will need and in what program(s). You want to start a search for the dates you will need in advance. If you know you are going to Timbuktu in February of 2020, then you begin searching for award flights in March of 2018 (330~ days out) that depart to Timbuktu in February 2019. This will give you a rough idea of how many miles you will need and in what program. You also have about a year to get those miles in place for March of 2019 to book your February 2020 award flights.
I knew well in advance that flights for our Antarctica trip would likely be at the Anytime level and that saver levels would be highly unlikely. This is actually the first time I've EVER booked AA flights at the Anytime level. I planned for blowing through about 800,000 AA miles for this trip and made sure to have a cushion in case of a devaluation as well as a plan B which was AA gift cards and UR points. Silver lining was that it only cost me 690K AA miles. DH and I will each get back 10K miles since we have the Barclay Aviator business cards and AA Citi cards. It was a better value to us use our AA miles over UR points for this. Having depleted our 1,000,000 + AA miles balance we will be focusing on the AA shopping portal for the foreseeable future to beef it back up. Hope this helps.