Hi there OP!
I have read your comments through the thread, but not the whole thing. My kids are 6 & 2. I work full time compressed over 4 days and quite a lot of travel, and lots of extra work/meetings from home. My DH is in the military and away a lot, or has very early/late duty times, so we don’t have anything like a schedule.
I don’t exercise nearly as much as I like (pre kids, I played my sport quite seriously, usually 5-6 days a week).
Right now, I run when I can, by myself - straight out the door and do 5k or 7k. I cycle when I can, too.
I prioritise my sport (a team game) and make a lot of sacrifices to get to it, at least two regular days a week in winter (I have been lucky enough to enlist MIL when DH is overseas, or I take the kids along and ask the team to help).
Recently I started horse riding again, which I love - every other week, at a lesson, booked for when DH is around.
And when I travel overseas, I take kit with me and run/swim/stretch or whatever I can do.
I am not fit, or fast, or slim, but I work really hard to keep up to it all! But the one thing I am doing right now which may particularly help you is HIIT training. I’m following the Body Coach 6 week training session. 4 times a week, in the evening, after the kids are in bed, I go out on the patio with a yoga mat and YouTube. The sessions are 15-25 minutes, no equipment. And I do what the Body Coach tells me. You may struggle a bit if you are carrying an injury, but there are modifications to reduce the impact.
I also do yoga with DVDs, or just ten minutes in my normal clothes doing my favourite stretches. With yoga, you don’t even need to get changed, or sweaty (which is sometimes the hurdle for me).
It is possible, I promise. And you’ll get endorphins and sleep better with regular exercise. Do try HIIT!