How to handle pressure from collaborators when life circumstances make it impossible to get work done?
I've been on both sides of your dilemma, both as the slow and unreliable collaborator and as the person dealing with a slow and unreliable collaborator, and I deeply understand your dilemma and your pain.

The first thing to understand is that this is not an unusual situation, and you aren't a bad person for having it happen to you.  In fact, this is a fairly normal situation of life for most people pursuing post-Ph.D. research careers.  The basic problem is this: collaborations, grant proposals, professional service, and many other aspects of a scientific career all have long time horizons and high uncertainty associated with them.  That means that you often have to make a time commitment without really knowing how large it will be or what it will be competing with six months, one year, three years down the line.  What that means is that, no matter what approach you take to personal time management, you are likely to be frequently faced either with significant unfunded gaps, with majorly overloaded periods, or even both in oscillation.

At the beginning of a Ph.D., most people do not have this problem, because they have precisely one research project, and they can work consistently on just that project and their courses (and maybe TAing).  This situation sometimes persists to the end of a Ph.D. or through a postdoc.  As a person moves towards independence of a researcher, however, they start to have more opportunities and more choices, and are faced with this dilemma of planning under long-term uncertainty.

Unfortunately, under the current organization of academia at least, there seem to really be only two solutions to this problem:


Be low-ambition as a researcher, ensuring that you are never overcommitted but greatly decreasing your chances of obtaining a tenure-track position at a strong research institution, or of making tenure once you have one.
Triage and honesty.


The best advice that I have ever seen about the second path is in this blog post: "The only "balance" is in choosing which ball you are going to let drop today and deciding not to drop the same ball repeatedly."

Now, as to what to do: first, you need to be honest with your collaborators about the change in your life circumstances.  Having a child radically changes your life and your available time: as a parent of a 2-year old, I have about 20 hours/week less for work than I used to, and that's despite making compromises I'm not entirely comfortable with in the amount of time I spend with my daughter.  Science will expand to fill the time allotted to it, so the first thing you need to do is to decide how much time you are setting aside for family, and guard that time with your life.  Practice saying things like, "I'm sorry, I can't meet with you then, I have to be parenting then," and "I'm sorry, but this weekend I have family commitments." and spend the time you need to spend with your partner and your child.  Second, you also need to take care of yourself: if you aren't sleeping well and eating healthily, your research productivity will drop even as you spend more time working and your stress level rises.

Now, if you do this, it means that you are not going to be able to follow through on all of your commitments to your collaborators.  There simply aren't enough hours in the day.  And that means you're going to have to triage.  In my experience, the most important thing here is honesty.  In my collaborations, I can plan how to manage a collaboration in which somebody tells me, "I'm going to need to put this on the back burner for a couple months," or "I'm going to be able to spend only 4 hours a week on this," but I can't plan how to manage a collaboration with somebody who tries to conceal their overcommitment or who repeatedly promises things that they cannot deliver.

You need to figure out your priorities for investment on your projects, not for all time, but at least for a few months.  You can reassess afterwards.  Talk it over with your advisor, to make sure they're OK with it, and then let your other collaborators know that you've had to reassess your commitments and what they should expect from you.  Be explicit about the fact that it is becoming a parent that is driving this change: any collaborator that is worth continuing to work with will understand and respect this fact and be willing to work with you on this.  Anybody who doesn't is somebody that you want to be very cautious about continuing work with in any case.

And make sure that you have enough space to enjoy time with both your child and your partner.  Our time in life is far too short and precious as it is.