I'm gobsmacked at this thread.
OP - yes, children are generally better off not being in full time nursery until they are over 3 years. Unless the parent/carers are abusive or neglectful.
You don't have to put your kids in full time nursery. you can do it part time, or look into using a childminder, again, part time. It might give you enough of a break to just breathe and get yourself back together again.
You write about your children as if you don't like them very much. We all have moments like that - but I don't believe that your kids whinge ALL DAY. No kids do. I think you are globalising your feelings and only seeing the bad in them.
If you decide they are just whinging, and therefore you can legitimately put headphones in and ignore them (!) - that's really not going to end well.
It will be a few years before both of your children play independently enough for you to get on with work uninterrupted for a meaningful length of time. So you do need to think about this problem of working with small children at home now.
I am self employed, work from home full time and my DS has not gone into nursery either (he is 2 and a half now). But I only have one child, my partner also works from home and is willing to share parenting duties equally, I employ a cleaner once a week AND we have approx 9 hours spread across the week from a very local nanny.
and I would still say that is manageable but tiring! and I struggle to keep on top of shopping and laundry and cooking. FWIW, i also have a history of depression, although am well atm.
You do need to make changes but turning up the music and leaving your children to cry is not the way forward.