I'm not in your position because my kids are almost 4 years apart. However, I have a 16-month old, and even that is very challenging with another child around.
First, I would say have nope, because in a couple more months you will have the nap an dnightime sleeping under control!!!
Second, pick a room maybe near the kitchen, childproof and barricade, set uup a pack n play or 2, and be prepared to spend you day there. Pack a basket with water for yourself, a sippy cup and snacks for the toddler, etc. Don't expect, especially in month 1, to be supermom doing it all.Position your self at child level on the floor and just give them lots of attention. It keeps the mess from spreading, and limits the time you have to be up and away from the kids and the toddler can keep an eye on you.
I don't believe in timeouts, I don't think they are very effective or educational - I think they are punative. In our home, we redirect, remove toys, or stop an activity that is causing problems. And even in cases where timeouts seem to help, I think 15 months is too young. However, if you need your toddler to be safe and secure for a few minutes, by all means stick him in the high chair or the pack n play or crib. I think 5 minutes of crying is better than the chaos of a loose toddler.
Also, have you thought about getting a mother's helper. Since it's summer, you could find a teenager or college student to help. Have her come over at toddler nap time, so you can get him down for some sleep. I think if you can keep his sleep schedule on track, that will help you considerably. She could just be an extra set of hands and a playmate, not a babysitter. I was a mother's helper when I was 12-13 and basically just played with the baby after school so Mom could make dinner.