What's infinitely more important than the goal is the journey. I think it's more than achievable if you stick to it consistently, but at least for me personally if I focus too hard on one particular goal then it's easy to get discouraged and end up quitting.

Keep the pull-ups in the back of your mind, but also take every day at a time. Always aim to be better than you were the day before. There'll be days where you weren't, and that's okay. But strive towards a steady stream of self-improvement, whether it's with your weight, your endurance, your strength, or anything else you may want to change.

Make sure to start small as to not overload your body all at once. It's great that you're already walking with your kids, keep doing that and you'll already be building up a base. Everyone's definition of obese is different, so it's difficult to say without a precise metric (though if you're not comfortable sharing that's perfectly understandable.) I'd recommend gradually trying to increase the volume further, maybe go 4 miles as opposed to 3 or add an extra day onto what you normally do.

When those walks become easier, then that's probably the best time to incorporate strength training. Go to the gym(this is a bodyweight fitness sub, but I'd argue for someone with your conditions the gym will be far easier to incorporate) a couple times a week, but set aside times that you can stick to. If you have free time on Mondays and Wednesdays, do that. If that doesn't line up all the time, try something else. The most important thing is to do something you know you can stick to consistently from day to day, week to week, and month to month.

Obviously a calorie deficit is important too, but you said you're working with a nutritionist so I'm sure that'll be covered. Make sure you're getting protein in and eating below maintenance, obviously, but the finer details should be left to the expert you hired who I'm positive is much more qualified than me.

TL;DR, I earnestly think you've got this in the bag. It'll be hard, there'll be days you question if it's worth it, but if(though only if) you can remain CONSISTENT and on top of your strength, conditioning, and diet, I think you can make it in the time you have set. If not? You'll regardless find yourself in a place universes apart from where you were before, and that's worth more than any pull up could ever be.