Pushchairs. A pushchair is a chair on wheels designed to be pushed. It is a generic term that could include buggies. But generally suggests more padding. And so is suitable for older babies, who are able to sit. But not able to walk. A modern pushchair with three wheels. But the number of wheels is not important.
The baby buggy initially refers to a particular brand: The folding chair with wheels designed by Andrew Maclaren in the mid-1960s. The original buggies were lightweight and no cushioning. The name suggests the “Beach buggy”. They are lightweight and can go anywhere. They are suitable for toddlers. Modern buggies may have more padding, but the key idea is that they fold up and are portable. This is the classic 1960s buggy, with its striped seat.
Many modern pushchairs are convertible: they can be configured as a pram for young babies, then converted into a pushchair when the baby can sit, and they can fold like a buggy. And so there is overlap in meaning, and words may be used imprecisely to mean “the thing I push the baby in”. So, for example, you may hear people using buggy for a folding pram.
Other terms: Stroller and Baby Carriage are not used in British English (except as an Americanism, several shops seem to be using the term stroller now, so it may see greater use in the future). Wheelchair is a chair on wheels designed to be moved by a disabled user.