If your wife is commuting to NYC, Hoboken and Jersey City have better public transit access into Manhattan via PATH which runs 24/7 and during rush hour, trains come like every 4 minutes. Union City and Weehawken require buses and going through the Lincoln Tunnel. Therefore, I'd vote JC or Hoboken for the commuting aspect.
Hoboken is very expensive, though. It's great for young families with young children, but very expensive. JC isn't much cheaper, but I feel like you do get more for your money. Just be careful because away from Downtown, things can get sketchy. Journal Square neighborhood is picking up a lot. The Heights is very nice, but doesn't have the ease of Journal Square, Grove St, Exchange Pl, or Newport PATH (subway) stations.
If your wife is cool commuting via bus into NYC, I'd say check out Jersey City Heights. More bang for your buck than Hoboken and Downtown JC. Safer than Union City, though Union City isn't bad.
I have no idea about public schools, but I know JC is putting lots of money into the public schools so as to retain the young families moving out of NYC, and keeping them moving further into the burbs. If you can't find a good public school in JC or Union City, you'll at least be saving money on housing to put into private programs. Also, Hoboken seems a bit maxed out to me. I think you'd be making a better investment buying in The Heights or Union City or Weehawken.
Here are some great articles on The Heights. It's a really hot market right now for housing, and lots of new restaurants are opening there. Browse through this site and check out the other articles on Union City and Weehawken as well.
https://jerseydigs.com/tag/the-heights/