Ah, that's quite a lot of extra pieces to this.
Firstly, well done on going to your GP – hopefully the antidepressants will have you feeling better very soon. Are you getting a chance to regularly have a break from both children and do something for you? Read a book, have a bath, go for a walk, whatever it is that will recharge your energy and patience levels a bit? A newborn and a preschooler can mean a lot of noise and a lot of physical contact for mum. Are you managing okay with that, or are either or both (or some other practical or physical thing) using up a lot of your coping ability?
I've been there with a DH struggling with his mental health, struggling to cope with normal small child behaviour, and it really straining their relationship and the general atmosphere in the house. It's horrible, and very hard to know how to balance encouraging him to seek help and to continue to be an active parent with managing the effect on their relationship and protecting your child from unreasonable expectations. No easy solutions, but a handhold, and confirmation that you're finding it very, very hard because it is very, very hard.
There are some groups aimed specifically at men that he might find more accessible. Andy's Man Club, I'm sure there are others. Can you talk to your health visitor about it? They may know of local resources he could access. Beyond that, as decent sleep and healthy food as both of you can manage, fresh air, natural daylight, exercise, and some time to do something he enjoys (and the same for you!). It won't solve a serious mental health problem, but can help support and stabilise things.
With your three year old, no, mums absolutely don't always know what to do. We just have to give the impression that we have a plan and have it all under control. I've had a couple moments just today of standing there, mind spinning, trying to figure out how to juggle the competing needs of two small people or how to give a logical, linked consequence for misbehaviour, close enough in time to be meaningful for a four year old, and ultimately getting me the behaviour I need. Probably not even my husband realised how much I was winging it.
And you can be gentle and loving, and hold the boundary. My four year old fought bedtime yesterday – she'd played up getting ready for bed, and so lost the chance to go downstairs for a few minutes of TV before stories and songs. I sat on her bed, stroked her hair, told her that I understood that she was upset about not getting to go downstairs, but I kept her in bed. (I would have physically put her back in bed if necessary, and have when she was younger.)
Overall, I tend to say what I need them to do, say I'm going to count to three, then I'm going to choose for them/do it for them/carry them/whatever, and I stick to it. (Our standard bedtime antics one is that if you aren't getting undressed/cleaning your teeth/etc. by the time I count to three, you must be too tired to listen and need to go straight to bed, i.e., no TV treat. It usually, but not always, works.) However, you can often avoid needing to get to that point by giving choices (teeth or pyjamas first) or being silly about it (are you hopping/crawling/flying/etc up the stairs has prevented many a bedtime strop here). And then praise all the little things, you want her hearing much more positive than negative. Would your husband be open to trying that, catching the good moments, that she's put away her shoes, or worked really hard on building a tower, or whatever?
That's an essay, but I hope there's something in there that's helpful, if only sympathy and confirmation that you've got a lot to deal with – anyone would find it hard.