I have travelled with a pushchair with BA. They don't charge you extra for the pushchair or the car seat. You can give the pushchair just before you board in the plane and they put it with the rest of the luggage. The car seat they took with the rest of the luggage. So you don't have to buy another pushchair if you don't want to yet.
My petite star zia has just completed its first trip by air, short haul on British Airways.
On the way out the flight wasn't very busy, and after checking with the crew we were allowed to take it on board. Folded up into its travel bag, the zia easily fitted sideways into an overhead bin on the narrowbody aircraft - not very space efficient if there are rollerbags to try and fit into the bin too, but plenty of room for other bags.
Coming home from Heathrow the person at the gate was determined not let us take the pushchair on board, despite the fact that the flight wasn't that busy, so it went in the hold in its travel bag. Irritatingly there were a couple of completely empty overhead bins near our seats that the pushchair could easily have gone in. Anyway, didn't really matter because I had the sling assuming that it would have to go in the hold, but it was the principle that annoyed me! Pushchair arrived on the baggage belt undamaged, so I'm happy.
Next time I might make sure the pushchair is in its travel bag before we arrive at the gate, so an overzealous person doesn't see what it is and make us check it in if the flight isn't busy.