In the UK, are the words 'pram' and 'pushchair' interchangeable?

No. A pram is large, cumbersome and less commonly seen now, perhaps because of the inconveniance of travelling with it, since it does not fold up. It is a bowl shaped container on 4 wheels. The baby always faces the mother. There is a hood which may be raised over the baby's head and the handle is at the foot end.

A pushchair covers a multitude of different types including the baby buggy.The child may face the mother or face forwards, which I much preferred as a child, The child is much freer to move around and can see more of it's surroundings. The main benefit is ease of travelling as a pushchair can easily betaken onto a bus or folded and put in the car.

They didn't use to be, but now they probably are. A baby carriage is for small babies who can't sit up yet, and are wheeled about lying down. The modern stroller can recline, so carriages are hardly used any more. Pram is short for perambulator, which used to be a synonym of baby carriage. A pushchair used to be a synonym of stroller. The word pram just transferred over to the new realities of baby wheeled vehicles. Now there are even jogging strollers, with shock absorbers to make the ride smoother.

Yes, though pram is far far more common. Still, "carriage" is well known. "John and Mary, sitting in a tree, K.I.S.S.I.N.G. First comes love, then comes marriage, then comes Junior in a pram" is kind of clunky.