I would imagine the dimensions of the carseats, even the smallest ones for infants, will far exceed BA's hand luggage maximums. Even leaving aside whether or not BA would allow something other than a suitcase or similar, you would fail on that alone - and given this isn't even close, you can be pretty sure the despatchers are going to notice.

On my last few short-haul flights I've seen enforcement of hand baggage in respect of forcing hold checking range from looking to stringent enforcement. In the summer months, with heavily-loaded planes, this is going to get even more attention. In short, I don't fancy your chances - and indeed, you may well find that even the security scanners can't take the seats and you'll be sent back to the check-in desks.

So by all means try, but be strongly prepared for it not to work.