They should give you the stroller when you exit the plane regardless if you have a connecting flight or not.
Obviously you if you gate-check the stroller
Here is what I got as email reply from BA:
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Dear ....
I am sorry we have been unable to reply to you sooner.
Usually you can keep the pushchair (stroller) beyond the check in desk area, if you want to. Our check in staff will provide you with a baggage tag to be fastened onto the pushchair.
Upon boarding the aircraft, the pushchair should be handed-over to a member of the British Airways team so it can be stored in the aircraft hold.
Upon arrival at your destination, you will find the pushchair at the aircraft side, if this is not possible, it will be delivered to the baggage arrivals hall.
Further, London's Heathrow and Gatwick airports have already advised us that, due to the volume of flights and passengers that use the airport, they are unable to deliver children's pushchairs at the aircraft side. It must be collected from the baggage arrivals hall.
This may also be the case at other airports and we advise passengers to check with the airport they are travelling from as we cannot guarantee that buggies can be taken beyond check in at all airports.
For your information, infants who have not reached their 2nd birthday will be permitted a baggage allowance of one piece and will be permitted an additional fully collapsible pushchair, and a car seat.
I hope the above information is useful to you.