Baggage (checked) allowance for infant is 1 x 23Kg bag + stroller + car seat. You can definitely check the stroller at the gate but not sure why you'd check the seat at the gate? Anyway it can go in.

BA accepts CARES but with a few caveats - can't use in emergency exits or rows either side and can't be used on WT+/CW seats etc. Whether all cabin crew know what CARES is is another matter and you may need to politely ask for the senior cabin crew member to help.

I have never used a car seat onboard but was told by a purser last year that BA do allow them if they conform to certain standards but couldn't tell me what! I believe the lap strap has to pass completely between the back and the car seat itself and that 2 part seats are not allowed. Also the harness has to be 5 point. Obviously the car seat has to point the same way as the aircraft seat. Your biggest problem will be getting the seat onboard if you haven't bought a seat for it - by the time you have negotiated for an empty seat they'd have taken it off you!

On some aircraft (particularly SH) there are only certain rows that infants can be seated and you may get moved at check-in because of this. I think it's to do with O2 masks. i.e. you can't have 3 infants all in the same row!

Have you considered asking for a bassinet location for the ORD-LHR sector? Child too big for it? That's a long sector without one! They do a cot for say up to 6 months and an adjustable seat for older infants. Only on long haul though.

The official BA line is no strollers onboard. However I think you will find, in practice, that the crew will be fine with you taking a fully collapsible stroller onboard with you IF there is room on the flight and it fits in the overhead bin. However IME the US ground staff (read stroller police) will have it off you whatever you say unless you shove it up your jumper. If you can find a way past the stroller police then you stand a chance of having it at LHR. YMMV.

Finally I can only say that I have just done 2 trips to the US with a 12-14 month old and cannot speak more highly of the BA crew onboard. They were simply great and I am sure you will be looked after. Unless of course you are unfortunate enough to get a dinosaur assigned to your cabin who doesn't know what a kid is

Good luck with the trip