My heart goes out to you!
The lack of any time at all is just unbelievable. My DD didn't sleep either - always wanted to be up in arms and wailed if left alone... I just used to put her down on some towels in the bathroom when I had a shower - she screamed with the noise, but I had to wash!!!
It is hard going, and my DD didn't change until she could sit up at 7 months. Thinking back, I wish I had put up black out blinds for her and had a firm bean bag to plop her in. we got a mobile which wound up and it gave me about 2 minutes.
what really used to bug me was that all my friends who had babes at the same time had no problem putting their LOs down, and were full of advice like - just put her down, she'll drop off......
I have to say I was more than a little bit relieved when their second babies turned out like my DD (she's an only BTW for obvious reasons!!) only then did they understand why I was so frustrated and wrecked.
So I feel for you and hope some of the suggestions here are useful.
If you can try and get someone to give your baby a walk, and get some down time and a long bath / facial for yourself. It's essential to recharge your batteries and just remember who you are again.
My DD is an amazing little 3 and a half year old now - really tuned in to people and their emotions, and very empathetic - so all the 'face time' we had was just part of her personality. She really needed to look at me and figure out my face and be held when she was a baby. Now I'm glad I just gave in and accepted that she was the way she was /is, as she's just such good company now and very much a people person with lots of little friends.
HTH and good luck.