Haddock, please do speak about it to the HV, and also have your cough/sore throat, etc looked at, maybe you need ABs.
Having 2 children under 2 feels like a car crash every day. I remember being shell shocked for the first few weeks, and my firstborn was 2.3 at the time.
She also talked a lot about the baby screaming. I bf both, and felt more able to bond that way. I think bf is a great way of being close to them, and the skin-to-skin contact helps calm them down. But I don't think bf to the point of exhaustion will help you.
Try to figure out what will help you. Little things, big things. Even silly things, everything.
Your mother&sister are out of order telling you ds2 was a mistake and not helping you at all. Unfortunately I also had no help with my 3, my mother just wasn't interested, my sister lives 300 miles away.
You may feel ds2 was a mistake, but then you're not yourself at the moment. Don't start to feel guilty for thinking and saying it, I'm sure many many mums have thought the same. Because you're feeling almost pushed over the edge. Sleep deprivation is torture. Your only aim should be to get as much sleep as possible.
Double buggy and let them both sleep in there together while you crash on the sofa sounds a very good idea, as was mentioned upthread.
While ds1 is at nursery, put ds2 in the buggy, and as soon as he drops off to sleep, you sleep.
I have seen your list for the week, looks like a plan. Meeting other mums is important, imo. You don't have to find a lifelong friend (although I did find them one on a playground and one in an nct newborn group) but just meeting other mums with the same stuff going on is vital, so you don't feel a failure! It's normal!
Hope you have your appointment today.
Btw I'm up because my 4yo about-to-start-school-tomorrow dd2 woke up, I'll go back to sleep soon. My first 2 are 12 and 14, you see, I needed an 8 year gap but then had another one.
I had pnd with my 3rd (due to health complications and scares just after she was born) and probably didn't bond with her until she was a toddler. I look back at that time now and have stopped feeling guilty.
Please look after yourself, your health is most important right now. Get as much help as you can. If you cannot afford a cleaner, try to get a babysitter for ds2 on the days he's not at nursery, maybe in the afternoon, so you have a chance to sleep.
Have a
and try to get some more sleep.