The question as always comes down to "make a difference" and "important" and other terms which imply that one is going to run a kind of hypothetical simulation in which history might have been different if you tweak a few variables. And for obvious reasons that is rather hard to do convincingly!

Obviously there are very few people who one can point to and say, "if this person hadn't arrived, then the Allies wouldn't have been able to do X!" That isn't really the nature of scientific and technological development, despite for it making good fiction, where a single person saves the day, etc.

That being said, there are a few of these emigres who had an outsized influence, to the degree that it is possible to imagine that they would not have been so easily replaced. Enrico Fermi and John Von Neumann, for example, both made contributions to the American nuclear and defense programs that are in league with that level of support. They were both considered by their (very smart) peers to be an entirely different level of genius in their respective fields of expertise.

Programmatically, one can point to places where Jewish emigres had a very large influence. The most obvious is on the atomic bomb question. In both the US and the UK, it was Jewish emigres who pushed for work on this topic, to a degree that the "native" American and British scientists thought was premature. Notably Leo Szilard in the USA (who enlisted Einstein to write to FDR to start government funding into fission research), and Otto Frisch and Rudolph Peierls (note to self: this is the first time I've spelled his name right on the first try) in the UK, who authored the Frisch-Peierls Report on the feasibility of an all-enriched uranium bomb, were important in getting these programs off the ground when they did. In the case of Szilard, the committee that FDR created might not have been enough to launch a full weapons program if not for a British report, spurred by the work of Frisch and Peierls, on the feasibility of a weapon, so one might be able to say that the Manhattan Project would have been successful without Szilard or Einstein (but it would have been different, in any event). But without Frisch-Peierls, you don't really get a workable bomb in time for the end of the war; it would have just delayed things too much.

I sometimes get asked, by half-embarrassed colleagues, why there are so many Jewish people involved in the bomb work in particular. And part of that is the standard answer about why there ended up being so many Jewish people in theoretical physics in the 1930s — because they were excluded from experimental physics on the Continent, which was seen as the "real work" of physics. But part of the answer is that these emigres in particular were the most likely to assume the worst when it came to the Nazis' capabilities and intent. American-born and British-born physicists were more inclined to say, well, the odds are very long that atomic bombs will be feasible, we should concentrate on more "safe" bets. The Jewish refugees were more likely to say, if there is any chance that the Nazis will end up with nuclear weapons, we need to beat them, because they have very good people over there, and thoroughly malicious intent. Their fear was as potent a factor as their scientific talents.

Whether the Germans could have made a bomb in time for it to matter is a different question (and you can find a lot on in the FAQ). The short answer is that the main obstacle to a German bomb was not that they did not have competent people, but that they did not invest in making a bomb. You need both competence and investment. The Germans did not fear an American or British bomb as much as the Americans and British feared a German bomb — the role of fear is again crucial here, and the fact that there was an asymmetry is part of why the Germans got nowhere (because they didn't invest in a bomb program) and the Americans built one (because they did). The Germans did have many brilliant scientists... but they were assigned to other tasks, like the V-1s and V-2s.

For the Manhattan Project, I find it an easier claim to make that if, somehow, the Manhattan Project started, it could have been likely successful without the participation of the refugee scientists. It might have made some errors or gone in some wrong directions, or at least different directions. They were involved at very high levels in the technical decisions and those do matter. But the US had a large "native" talent pool as well, and its industrial capabilities were most of the reason it accomplished its task in the given amount of time. That is not to say that those who participated did not make important contributions, it is only to say that in our hypothetical counterhistorical world, someone else would have had to have risen to the challenge of making those same contributions, and it is not impossible to imagine that they would, with the exception of the fews cases already noted (Fermi, von Neumann, maybe a few others).

Obviously all of the above should be taken with a grain of salt, as it is interpretive and impressionistic rather than a real historical "calculation."