Assuming they just filled the "gaps" in the current schedule to SFO, LAX and ORD, not to mention BOS and IAD would be 4-5 aircraft alone. The 18 (or is it 23, Im not quite clear on that) Super-Hi-J 744s will be the last to leave the fleet, but not before 2022, based on current public plans. The current Lo-J aircraft will be replaced one-for-one by the A350, the configuration is unknown yet, but expected to be similar to the Lo-J 744s, maybe with a small loss of Y seats. These are expected to be seen on CPT/DEN/LAS/off-peak JFK, etc.
BA needs to make a decision on what will replace these 744s - it might be A32x for all we know, but frame-for-frame they will be replaced. BA won't give up their slots. Clearly BA is shopping for wide bodies, around 18-20 of them. I could imagine an order for 8-12 A380s and a similar number of A350s.
Right now there are 36 744s, 18 with 86 J seats, recently refurbished, and 18 with 52 J seats. BA's most recent announcement (November 2017) was that 24 of the 744s would leave the fleet by the end of 2022, being replaced by A350-1000s and 787-10s that are already on order. There are currently no orders for delivery 2023 or later.
The 744 is to leave the fleet altogether by February 2024. So they need to order 12 planes to be delivered during 2023. Options exist for 787s (type unspecified), A388s, and A350s (in large numbers, held by IAG rather than BA). If they make a reasonably quick decision, they should be able to convert options on any, or a mix, of these types and get them delivered in 2023. The A388s would require an increased production rate to manage it if 12 were ordered all for delivery in the same year, but this seems entirely possible.
BA will also do a light refurb on a few of the 18 52J unrefurbished 744s, but not all of them. We don;t know if the configuration will change on these, or not.
A350-1000 and 787-10 deliveries are phased between 2019 and 2022, so some 744s will disappear each year, with some of the earliest maybe even leaving at the end of this summer's timetable.