I'm still gonna have to say that I didn't see it. Granted I've only played through any route once, but the first indicator I really saw was her angrily hitting the ground. I can't quite remember if Hisao thought she was acting odd just before Akira took him aside, but even so. Lilly seems to go out of her way to present herself as being self-assured and capable... assuming she's troubled about something based on a couple odd quirks here and there doesn't leap to the idea that something major is automatically happening.
I remember reading some joke post online somewhere. Hers and his journals, or something.
Hers goes first, and what it says is roughly a few paragraphs of "he's been distant for a couple days now, I try to talk to him but he says nothing's wrong, he's a lot more quiet lately, and he always acts as if his mind is on something else. I wonder if he's wandering, or thinking about it? Is this the first sign? Is our relationship coming to a close? Oh, God, I don't want this to be happening..."
His is "still can't figure out why the motorbike won't start..."
Could he have pushed her a bit more when she started showing signs? Sure, but he's got no reason to think that whatever's bugging her is as serious as it is. Hell, he's got plenty of reason to assume that he's what's bugging her, or more specifically, her (justified) worry over his health is what's bugging her. How many attacks did he have right in front of her by then, anyway?
The guy doesn't have the experience or the insight to know that she's struggling with such a major decision, and frankly, I think that if something that important comes up, and you have someone important in your life that it will affect like that, it's on you to open up to them about it, not them to assume something's wrong and ask you.
There were some signs in addition to her sudden angry outburst. Hisao notes earlier that Lilly used to have very clear plans for her future, but now whenever graduation or her future comes up she starts evading the subject. He can feel some distance between them that wasn't there before. Sure, it's subtle but most stuff about Lilly is subtle. Hisao identified Lilly's outburst as a red flag, but when she recovered her lady facade, he decided against pressuring her. But yeah, this was as much Lilly's fault for not confiding in him as it was Hisao's fault for being too passive.
The major place where Hisao dropped the ball, however, was when he confronted Lilly with the knowledge about the summons. Lilly appeared very unsure about the whole thing and even said "My parents dearly want me there and Akira's going too, so....". In other words, she's not leaving because she wants to, but because her decision is made completely with others in mind. That's where Hisao should have chipped in and asked her what it was that SHE wanted. But by then, he was already so used to just going along with Lilly's decisions without questioning them that he didn't think to challenge her choice.