Mom of 2 here. You have my deepest sympathy. This can be a really hard adjustment, and you are currently in the thick of survival mode. That's normal, and it also sucks.
Balancing work, parenting a newborn, and exercise is hard. You can't excel at all of these things. Some of these goals are going to have to be shelved for a little while.
Work:
At 12 days in, I'm sorry you're back to work already. Can you focus just on the essentials? Do just enough to get by and have confidence that you'll get your groove back eventually. Even better, could you cut your time back a bit? Take some leave? My spouse cut back to 1/2 time for month when our second was born, and it really helped with the adjustment.
Health:
Can you put the baby in a carrier or wrap or stroller and go for a long walk? That's how I got most of my post-partum exercise. Look, you're not going to be able to do long, hardcore workouts for a while. For weight gain, I found that the biggest help was keeping healthy food around the house. Realistically, you're going to stress-eat while you're overtired, so make sure there's not too much easily-accessible junk.
Parenting:
At this stage, sleep is key. Can you trade off to make sure that, over the course of the night/day, both you and your wife get some significant stretches of sleep? If your wife is breastfeeding, it can be a little more difficult for you to take over a feeding, but there are things you can do to ease things for her. You can be in charge of nighttime diaper changes for a part of the night, or all night. It sucks that you don't wake up to the baby's cries--that would drive me crazy--but there's really not much you can do about that. You'll just have to figure out what works for you, whether that's splitting up the night into shifts, or both getting up when the baby is up, or you getting up when your wife nudges you. Honestly, we did better when we moved the bassinet to an adjacent room at about 4 weeks. I simply could not sleep through all of the baby's grunts and coos, even with white noise.
Sleep when the baby sleeps. I'm sure everyone has told you this, but it's critical. Don't use that time to clean or cook or do a project unless absolutely necessary. Sleep. Let some things go around the house. Hire it out if you don't have anyone to help you.