No dog here yet (trying to talk DH around!) but to say, when a lot of people have DC1 they can't imagine managing to do anything other than just about get the baby and themselves washed and dressed and fed. I was like that with DC1, getting to something that started at 10am was a struggle, I achieved nothing all day. Now I've had DC2, I have to get DC1 to preschool so leave the house at 8am, so this time round, we are all up dressed, breakfasted and out by 8am, we manage to go out again to collect him. I have to manage DC1's other activities and DC2 just has to fit in.
What I'm trying to say is you just cope with what you have to do. For some people, the idea of going out every day for an hour or so's walk would be far too stressful, but if you have to do it, you just fit it in your schedule.
I would make sure you have a good off road buggy though and not just rely on slings, with DC1 my back was rather knackered after childbirth so it was nearly 9 weeks before I could comfortably wear him in a sling for more than 20 minutes. If you need to be able to go out off road before then, you'd need a suitable buggy.
(oh and DC1 would sleep in the buggy, I often walked 2 hours every day getting him a good nap, you could just arrange your dog walks for your baby's nap times, as long as you keep moving, it'll be fine - and you'll get the baby weight off fast!)