Biddy leaves the forge when Mrs. Joe dies. She was there to take care of Mrs. Joe, so she was no longer needed.
After Mrs. Joe dies, Pip asks Biddy if it will be difficult for her to stay. Even though she loves Joe, she can’t stay because it would be improper unless he asks her to marry him.
“Oh! I can't do so, Mr. Pip,” said Biddy, in a tone of regret, but still of quiet conviction. “I have been speaking to Mrs. Hubble, and I am going to her to-morrow. I hope we shall be able to take some care of Mr. Gargery, together, until he settles down.” (ch 35)
Biddy’s job was to take care of Mrs. Joe after she was attacked and became an invalid. When Mrs. Joe died, there was no reason for Biddy to remain. Pip was aware of this, but he was also sad. He saw Biddy as a friend, and she was a great comfort to Joe. With Biddy gone, Joe would be alone and lose the one good thing in his life. Although Pip was not aware of this, he was at least aware that he himself would miss Biddy.
In Pip’s selfishness, he always cared for Biddy on some level but was not aware of the real relationship between Joe and Biddy. It does not occur to him that the two of them, after living together for so long, might have developed feelings for each other.
Of course Joe does ask Biddy to marry him, and they have a child. Pip never marries.