Your recovery will suffer some in the short term. Most of your muscle strengthening and adaptations happen while you are asleep. However, babies do learn to sleep through the night fairly quickly. At most you are looking at a month of the kind of sleep patterns that would cause problems.
So the short answer is "yes" you can go to the gym, lift weights, etc. The longer answer is that you will have to keep an eye on your progress. If you can't get weight up you normally would (loss of performance due to overtraining, which in turn is due to insufficient rest), you'll probably need to deload a bit more often.
The best approach, if you can't muster enough rest to make any gains, is to maintain. Basically your workout sessions won't be going up in weight while your sleep is suffering, but you won't be going completely sedentary either.
Your child isn't going to keep you up at night forever. There are things you can do as a parent to help you get as much sleep as you can. In the first month, the baby won't be able to make it all night without waking up and crying. It's usually because of being hungry, needing to be changed, or they are uncomfortable. If you control the child's schedule instead of allowing the child to control yours, you can work the child's natural sleep pattern to mirror yours. Don't respond to simple cries for attention when it's time to sleep, but do respond to the big three issues (hungry/diaper change/hurt/sickness).
The bottom line is we do need sleep, so new parents need to learn the difference between something that requires attention and something that can slide. All too often, new parents jump at the slightest sound from the bassinet. The child hasn't even started crying yet and they are running to its side. The child's cries will be different, and you'll learn which ones need attention and which ones don't. Until then, when the baby cries, check all the important stuff, reassure the child without picking them up and go back to bed. Get some sleep. They will learn, and so will you.