What is Winston's biggest fear in meeting with the girl in Part 2 Chapter 1 of 1984?
In this chapter of 1984, Winston's biggest fear in meeting with the "girl" (Julia) is that she will not show up. This fear derives, in part, from practical reasons: it is against Party rules, for example, for Winston and Julia to have a relationship, so meeting up is filled with inherent dangers. Winston refers to this in the text:
"The physical difficulty of meeting was enormous. It was like trying to make a move at chess when you were already mated."
As such, Winston fears that she will not show up because she cannot avoid the eyes of Big Brother, specifically the hundreds of telescreens and hidden microphones which stand between them.
On a more personal level, however, Winston fears that Julia will not show up because she does not really love him. We see this when he is waiting for her in Victory Square:
"Again the terrible fear seized upon Winston. She was not coming, she had changed her mind!!"
This is hardly surprising since Winston and Julia do not really know each other and her love note came as a complete surprise.
So, for Winston, his biggest fear is that she will not show up. But Winston is not certain if this will be a result of a change of heart or Big Brother's ever-watchful eyes.
In this chapter, Winston has been trying to figure out how to get a chance to meet with the girl (we will later find out that her name is Julia). He no longer thinks that she is a spy for the Party (like he did before this chapter. However, he is still afraid when he is trying to meet her. His fear is that she might have changed her mind so that she is no longer interested in him. Orwell tells us
...but now a terrible fear had taken possession of him. A week had gone by since she had first approached him. She would have changed her mind, she must have changed her mind! It was impossible that this affair should end successfully; such things did not happen in real life.
There are two ways to interpret this. First, he might simply be afraid for his life. He might fear that she will betray him and he will be killed. However, you can also argue that he is experiencing a more normal fear of rejection and embarassment. He might just be worried about how stupid he'll feel if he gets "shot down."
So, his greatest fear is that the girl is no longer interested in him. He may be afraid for his life or he may be afraid of looking stupid, or both.