What Wolf Alpha said is very true, however I would like to add a few things.
A bachelor's degree in a relevant field is less important if you have good experience.
Although some doors will still be closed to you, if you have lots of good experience, many employers won't care. The problem is that most good opportunities to get good experience require a bachelor's degree.
A bachelor's degree in a relevant field will teach you useful skills that aren't covered in bootcamps or nano degrees.
There is a lot more to being a good software engineer than knowing how to code. This is true for other engineering fields as well. It is still possible to learn these skills if you are very disciplined and know what to look for via MOOCs, but this won't work for most people. In addition, even if you manage to learn the equivalent of a a bachelor's degree in a relevant field, you may have a hard time convincing people of it.