Hi Donna and welcome! For what it's worth, my opinion is below.
I've had experience of both pups and babies and have had both at the same time, but only foster pups for a couple of weeks at a time. Being completely honest, there is no way I'd have a pup and baby together permanently, personally I just couldn't have coped but maybe that's just me! That's why we decided to foster, to get our 'doggy fix' without the hassle of a dog long term with young children.
A puppy is incredibly hard work (I actually think they're harder than a baby - they get you up at night and need constant supervision like a baby but unlike a baby they also need lots of training, destroy everything in their path, and teeth all at the same time - with kids it comes in stages but with a pup it's all in one go!). Your son/daughter is probably at her easiest stage about now as she can't move and is presumably on 3 naps a day - it gets much harder when they're older and are on the move, need weaning, only sleep once a day, etc, etc. Ask yourself if you could deal with more than twice the work you've got now, because that's about the amount I think having a pup and an older baby is.
The pup would also need lots of training, is that something that you have time for? I know when my children slept it was the only time I got the washing, ironing, cleaning, etc, done, so wouldn't have wanted to have to spend their nap times training a pup in the garden. The result would have been a very chaotic house indeed but I should point out that I'm not the kind of person that can deal with that, I need it to be neat and tidy or I go crazy! So perhaps that's not as relevant for you?
Added to the prospect of a very bouncy adolescent dog and a toddler trying to learn to walk (speaking from experience of having my son knocked over by a foster dog constantly and being hurt several times - it took a good few months before he was happy with dogs again after that). And pups nip constantly, again, you'd have to be very wary as a pups teeth would easily break a baby's skin.
I think another thing to ask yourself is how would you do at least an hours walk each day with the dog and baby - ok at the moment when the baby can be in a sling or pushchair but not so easy when you've got a toddler that will be too heavy to carry, refuse to go in a pushchair and only walk a few hundred yards before giving up!! (again, speaking from experience trying to walk foster dogs with a toddler!). A springer is a dog that needs a lot of exercise so you need to factor that in somehow.
We're currently getting ready for our pup but now is the perfect time for us - my daughter starts school in the new year, my son starts pre-school, so I'll have time to give to a puppy, and both children are big enough to deal with bouncy nipping pups and go on long walks with us.
It's obviously a personal thing and a pup may suit your household very well but I've just put down a few things that I thought of, and a few of the reasons why we've waited until now to get a pup of our own. If you do go for it then perhaps a breed other than a springer might be better, something not as big (so won't knock a child over as much) and that doesn't need as much exercise. Our neighbours have a cavalier king charles spaniel that they take for a 15 min walk each day which is fine for him - a springer would go crazy on that I think!!
Or you could always offer to become a foster home? That would give you idea of the work involved and help you to see if you want a dog full time - plus it would help out a rescue.
I hope that the above makes a bit of sense. Sorry if it doesn't but I just 'brain dumped' and typed whatever came in to my head!! Best of luck making your decision.