I'm also an engineer self taught in math, it's doable to teach yourself; I have since published in a pure and computational algebraic topology journal as well as several publications in combinatronics. In addition I use advanced modern math a lot in my engineering research.
My first piece of advice is to start out by looking at math bachelors curricula and buy textbooks following their program, supplement with freely available notes. The /sci/ wiki (not sticky) is also really good.
First of all it's possible to jump ahead and learn from advanced texts, it's just very inefficient to do so. Real analysis is really easy to understand since it builds on simple single variable calculus you don't need more background. The exercises are harder and a far greater time investment which I would not recommend unless you want to eventually do math gradschool or something. Then you need abstract algebra. Finally a point set topology book. This is the bare minimum you need to even understand most math textbooks. Now, because you won't have much time to practise and you don't have access to professional help I would also recommend you grab the high school level ones. For example Visual Group Theory really helped me a lot, as engineers we tend to prefer geometric and computational interpretations. Focus on applied textbooks and geometric approaches to subjects (especially when you start studying bilinear algebras) as you can usually understand them easier.
One last piece of advice is that HoTT is almost more related to computer science than category and homotopy theory, so if your goal is to understand HoTT I would search for applied books written by computer scientists for most of the prerequisite subjects.
You can definitely make enough time in the day to learn it. Then again I only have a fiance and no kids so idk if it's possible in your case, but it's worth a shot.