The word away when it follows a verb as it does there expresses the idea of unrestrained repeated or ongoing action. When the actor is a sentient being, the idea can be that the actor is "totally immersed" in the activity, that is, not paying attention to what's going on around him or her. When the actor is non-sentient, the meaning is that it was doing the action without interruption, continuously.
They were chatting away.
The baby was babbling away in its crib.
The stew was simmering away on the stove.
To gossip is to relate tidbits of news to another person. To gossip away means either to relate a succession of tidbits about different subjects, tidbit after tidbit after tidbit, or to tell a fairly long story with many details. Usually the news relates to a local person or someone related to them; the story is transmitted by "hearsay".
P.S. Gossip can be about personal details that people would normally like to keep private (e.g. the couple down the street had a loud argument the other night and may be getting a divorce) or bits of trivial local news not of a private nature (the neighbor's cat just had kittens, the boy down the street who delivers the newspapers had his bicycle stolen). Since the story uses the word happily to describe Mrs Dursley's behavior, and the phrase gossiped away, the inference I draw is that she is relaying trivial bits of news that do not involve anything that people would want to be kept private. We would not normally expect to find the word away used to describe someone who is revealing news of a salacious nature; "juicy secrets" are not normally delivered in a free and unrestrained manner while one tends to a screaming baby.
