For starters, a little intro. I'm currently at UIUC in Nuclear Engineering, and I've accepted at UW-Madison, however if I chose based on which school I liked the most I would choose Michigan. Was looking at computational nuclear fuel cycle and reactor physics, so I can speak a bit about all three from both a school standpoint and research standpoint.
To be completely candid, I think you can't go wrong here. I work for Katy Huff at Illinois and will work for Paul Wilson at UW and I think they're two of the best professors and advisors you could ever find. Professor Tomasz Kozlowski at Illinois is also awesome. I don't know which country you're coming from but Illinois has a very high international population so if that's something you're interested in, that would probably be the best school. I don't know if you've gotten offers from specific professors at Michigan, but Professor Brian Kiedrowski, Professor Edward Larsen and Professor William Martin (the three of them offered me a joint offer) all seemed fantastic to me. So advisor wise... from the bottom of my heart, I mean it when I say you can't go wrong. I am a bit biased because I knew Katy Huff (Illinois) and Paul Wilson (Wisconsin) before applying to grad school, but all three were fantastic options.
So, I'll list some more info about the schools in general to help out as much as I can
Still being super candid, Michigan has the most money and offers the highest stipend (by a lot). They were practically throwing money at us during the visit weekend. UW is the lowest, lower than Illinois by about $200/mo
None of the three schools are located in large cities, but Madison is by far the largest. If you're looking for a small city feel, that's the way to go. Ann Arbor is in a good sized town, slightly larger than Champaign-Urbana area. Highest cost of living is probably Ann Arbor, then Madison (though not too far off, I think). Champaign is definitely the lowest, you can find apartments/houses not too far from campus in the $400-500 range without too much of a problem. For Illinois, most of the apartments are in Champaign (west of campus) and houses are in Urbana (east of campus). You can find either in both towns, but it's just overwhelmingly one or the other.
Now for super random stuff that just popped into my head.
If you care about sports/school pride, Michigan has the most, closely followed by UW-Madison. Illinois has some but you certainly have to search for it in the engineering community.
If you're looking to fly a lot (on a budget), Ann Arbor might be the best because it's pretty close (half hour-ish) from Detroit. Madison has a decent sized airport, but only Delta and Frontier fly from there (I think). Might be better off flying from Milwaukee (1.5 hours away by driving) or Chicago (2-2.5 hours). For Champaign, it's usually better to fly out of Chicago (2.5 hours) unless you're willing to pay a much higher price.
The ANS (American Nuclear Society) chapter at Illinois is really awesome, but doesn't get a large participation from grad students. UW Madison is also great, and might have a slightly larger level of graduate participation. University of Michigan has traditionally been not very active in ANS
Illinois department might be slightly in flux (administratively) next year because the department head is changing. Don't know who yet so can't say if it's going to be a good or bad change. However, it'll definitely be a change since we've had the same department head since... 1998
Wisconsin requires a lot of classes to get the PhD so you might be taking classes longer than you would be at other schools (that's what grad students told me), although I think 5 years is typical graduation time from all 3 schools
Sorry I just dropped a ton of info on you, but these were like my big 3 schools. I love Illinois, I'm suuper excited for Wisconsin, and I (kinda) wish the advisor was a better fit at Michigan because it was so nice and I fell in love with Ann Arbor, so I'm pretty passionate about all these schools. Feel free to ask me more questions!!!