How much did she contribute: she contributed the physical understanding (in part with Frisch) of nuclear fission, and many of the related calculations, along with her earlier work (in Berlin) of setting up the experimental apparatus, etc. It's a big contribution. Hahn's contribution was also large. If I were weighing them it might be Hahn #1, Meitner #2, and the others after that. Hahn certainly did not have the full understanding of what was going on by himself.
On the Prize: there were many factors involved in Meitner not winning the prize, and it has been hashed over by historians and other physicists quite a lot, especially since the Nobel records on the 1944 prize were opened several years back. Among the factors are:
She herself never lobbied for it, and often deferred to Hahn, and frequently distanced herself from some of the bigger implications (e.g. atomic bombs and technology).
Hahn himself was a credit-seeker and aggressively re-wrote the narrative of things to give himself even more prominence in them (and people listened to him, in part because Meitner did not put out her own "narrative").
While in Sweden (where she fled after Germany), she made some prominent enemies amongst the Swedish physicists (which may have involved sexism on their part), several of whom were prominent in the operations of the Nobel foundation.
There is an inherent problem of it being a 4-party discovery for a prize that can only give 3 awards. In theory you could imagine Hahn getting it for Chemistry and Meitner for Physics, or even a mix of Hahn/Strassman for Chemistry and Meitner/Frisch for Physics. Parceling out credit is always difficult for Prizes of this sort, and often controversial.
There were several opportunities and various people did lobby for her to get the prize over the years, but it just kept not happening. It wasn't obvious, including to Meitner, that it wasn't going to happen for several years. (There is no reason that hers and Hahn's would have had to be for the same year if they are different Prizes, e.g. Chemistry and Physics.) In the end other people edged her out, and we don't have the inside knowledge of all of those "edgings." What are the relative weights of the above factors? It's not easy to tell — again, we're not looking at some huge referendum by the "scientific community," we're looking at the sub-committee of the Nobel Prize foundation that is charged with looking at such things. And in such situations, a few people's opinions, assumptions, views, etc., can make a big difference. Meitner herself seems to have felt that the main factor was her alienation of her Swedish colleagues, but in retrospect a lot of historians have faulted Hahn for writing her out of the story (or delegating her to a lesser role in it) as one of the reasons that it didn't happen later.
The role of sexism of course is what a lot of people want to ask about. It is hard to put a number on that. Sexism is not just some force that either does or doesn't exist, like racism and classism and other prejudices, it is something that makes up entire world-views and systems and psychologies without necessarily being stated explicitly. It isn't a simple variable you can isolate from other things, in other words; it's part of the world these people lived and operated in, and it is indisputably still part of the world of science. It's hard to imagine that it didn't play a role in the many choices over the years that have led to the exclusion of women from the Nobel Prizes in Physics (only Curie and Goeppert-Mayer), but it's also not so easy as to say, "oh, in this case it was clearly primarily sexism that excluded them." It's part of the "mix," and can be subtle: Hahn's willingness to write her out of the story and to "demote" her work and contributions is perhaps a form of sexism, as is her Swedish (male) colleagues who thought she was "difficult." But it's not the kind of variable where you can say, "in the absence of sexism, this woman in science would have gotten the Prize" — I mean, maybe? Maybe not? In the absence of sexism, who knows what this woman in science would have done; it was part of her world. People rarely say, "I oppose this thing because I'm a sexist" (that's usually only how it works in movies that are trying to signal to use that the past was sexist) — sexism tends to inform their evaluation, behavior, and arguments in more subtle ways.
Anyway, for a very detailed discussion of the various factors involved, Ruth Lewin Simes biography of Meitner (Lise Meitner: A Life in Physics) is the standard resource and covers this in a lot of depth.