Catherine Morland fell first, but ...
... Henry Tilney was never far behind.
The following "analysis" was originally part of an reply to an ask, but I thought it might be interesting to add to the general Northanger Abbey tag.
Please keep in mind this is only my personal opinion, based on my own subjective reading of the novel.
There are, I think, at least to me, two primary ways to look at Northanger Abbey and interpret Henry's interactions with Catherine in Bath, considering the time, social rules and most importantly, how Austen's other heroes are portrayed in their behaviour towards women (as well as Austen's less honourable men.)
One option is, that Henry is a shameless flirt, more interested in his own entertainment than in Catherine's wellbeing and absolutely willing to risk both her heart and her reputation for his amusement, Eleanor's comfort and the general peace inside the family.
Because his flirtations from the get go, are shameless. He's very forward, not only thinly veiledly complimenting Catherine, but also fishing for compliments in the same manner, expressing an interest in getting to know her better after the very first dance. He's very teasing, very flirtatious and VERY charming. He's having fun, yes, but if he doesn't mean anything by it...
That'd be cruel. As well as very selfish and careless.
And allowing Catherine to trust him as much as she does in the James/Isabella/Frederick dilemma?
Singling her out as he does?
Not interfering with the plans of her visit to the abbey, STAYING there himself for most of the stay when he is very much single, Catherine very obviously interested in him, they have already spent so much time together and there is no other young gentleman in the party to warrant his stay at the abbey, rather than his own home? Yeah, that's just plain irresponsible if this was nothing but pure fun without feelings. If he only fell for her later, accidentally, if he had no romantic feelings for her during their stay in Bath.
Because it could tarnish her reputation and it would only be natural for Catherine to have high hopes and... ultimately it would break her heart.
If this were to be the case he'd be toying with her feelings and no better than, say, Henry Crawford. Worse even, because Catherine is young, innocent and yet unengaged.
Now the other option is that while Catherine fell first and fell openly, Henry always followed her right on her heels. Yes, the author states:
I must confess that his affection originated in nothing better than gratitude, or, in other words, that a persuasion of her partiality for him had been the only cause of giving her a serious thought.
But one, serious thought, so basically going from simple crush to "yes this will be my wife". Which is fair, because Catherine only really started to think seriously of Henry after James got engaged to Isabella. Before that she liked him very much but the thought of Henry becoming one day her husband, or even more so, the wish of him becoming her husband, had not yet entered her head. This does not contradict the possibility of a (intense) crush on both sides.
And two, with Catherine being Catherine, she was likely partial to him from the get go and was showing it from the first second they got introduced to each other. They danced one dance before Henry says what I'm going to quote next and I think one dance, so 30-60 minutes full of open admiration, is more than enough to develop a very healthy crush on someone ;)
So yes, I am convinced that Catherine had very much a crush on him, but so had he on her. He saw her admiration and he liked it and even more so, he liked her. We have this line:
“I danced with a very agreeable young man, introduced by Mr. King; had a great deal of conversation with him—seems a most extraordinary genius—hope I may know more of him. That, madam, is what I wish you to say.”
Now the most obvious part is the fishing for compliments, he really isn't subtle about that one, but there are two things, in my opinion, much more important.
"Hope I may know more of him" and "That [...] is what I wish you to say" He wants Catherine to think well of him. Henry doesn't just want compliments here, he expresses, and quite directly so, the wish for Catherine's esteem. Let's remember that this is about her journal, so something supposedly only she gets to read.
Thus even though Henry uses the word say, what he means is think. It's less about what she ought to write, and more a way for him to convey that he'd like her to like him, to think well of him and to hold him in high regard. Now Catherine should have answered this with a compliment to show that she understood what he was "hinting" at, or at least assure him that she would write exactly that, because of course we as readers know that it's in line with her feelings, but... Catherine didn't really get it, did she? Henry's not subtle at all, yet still too subtle for her.
But also, and even more important to me, Henry wishes that Catherine hopes to know more of him. Or, plainly: He wants to meet her again. He wants to know more of her. He's basically asking for a second date. He's interested in her and wants her to likewise be interested in him and he's pretty sure she is, but he wants her to assure him that this is, indeed, the case.
So I don't think it's that far fetched to say that Henry did, indeed, have a crush on her.
Wich is in line with his immediately asking her to dance when he sees her next (though he gets refused then) and later:
To escape, and, as she believed, so narrowly escape John Thorpe, and to be asked, so immediately on his joining her, asked by Mr. Tilney, as if he had sought her on purpose!—it did not appear to her that life could supply any greater felicity.
Why yes, Catherine. As if he had sought her on purpose indeed. Not that he ever would, because why would he but... I can't help but smile at how oblivious and HUMBLE Catherine is. Yes, Henry did seek her out on purpose, Henry likely only came there to dance with her. This is after the conversation with Eleanor, where she, when Catherine asks whether they'd be at the cotillion ball answers thus:
“Perhaps we—Yes, I think we certainly shall.”
There were no plans for it. The beginning of Eleanor's reply sounds almost as if the Tilneys had intended not to go. There's a very good chance Catherine's and Henry's mutual desire to spend time with each other was the main reason for the Tilneys to go. So one'd better hope he sought her out on purpose, yes.
With this, I think, it is safe to think that Catherine fell first, but Henry never let her get ahead too far of him. Who fell harder is hard to say ;)
For those interested in the full reply, it can be found here. The original question was inquiring after my thought process behind my fic "The Winner Takes Naff All", which is set on the evening after the prevented morning walk and explores a different, not entirely serious, route how things could have cleared up.