I think the variety of different advice here shows how different it can be for everyone - it’s not a one size fits all solution.
i think everyone struggles to find the time initially, regardless of how busy you are. Like you, I also walked the dog every day but wanted to start doing more. I considered where the pockets of spare time were in my day, and for me that’s lunchtime in the lighter months, which is when I started. I couldn’t get a decent workout in at the gym in one hour including travel time, so I started doing YouTube workouts. I started with get fit with Rick, who has plenty of free workouts which start from 10 minutes. It’s easier to find a ten minute window than commit to a full gym session! I find his workouts are really easy to start with and you can find slightly longer to move up to if/when you’re able. There’s something about the structure of them that I find really fun! I still do them now, but I also use grow with jo (also shorter workouts available and she does a few which incorporate weights too which you might like).
I can’t get on with Lucy Wyndham read but I’ve found I really enjoy walking workouts, so I’ve since also found Kat gates-buettner and bodykindfitness are good for these.
now it’s dark in the evenings, I walk the dog at lunchtime so do my workouts in the evenings, and I’m joining the gym again in January to start mixing gym workouts and YouTube workouts, I may start getting to the gym before work when the weathers a bit better, but that’s a way off!
I understand why people suggest running but for me, I hate running and would never do it, so c25k would never have worked for me. Japanese interval walking might be a good start point for you though, it’s basically a walking workout where the speed varies, I really liked it.
in terms of the treadmill, I like the idea of saving a certain tv show or audio book or whatever just for treadmill time, but what worked for me is putting a playlist together of good bouncy music and adjusting the tempo and incline to the beat. It’s more enjoyable, it makes you feel like you’re strutting along to the music and/or in a music video (!) and I’d rather work at a higher intensity committing to 3 songs (which soon becomes 4, then 5, etc) than 30 minutes at a steady boring pace.